To make a more concise response to the question, let us dictate the state of Asia (where Buddhism is present at the moment) at the point of the rise of Islam and also to analyze the extent to which Islam affected the decline of Buddhism in various regions.
1) The first point to address is the Buddhist world at the onset of the 7th century and into the middle ages as Islam began to compete with the varying Buddhist communities in Asia.
As a rule, Buddhism at the moment is primarily locked into Asia, circulating outward from Hindustan (especially the east and northwest) toward East Asia, Afghanistan, Central Asia and Southeast Asia and somewhat to a lesser degree toward the Indonesian islands of Sumatra, Java, Malays, Borneo, the Cham peoples and the various island strings that make up the 'Filipino' lands. Buddhism as a whole is strongest in these areas and outside of these realms, has little or no existence. Thus, Buddhism is virtually non existent in Arabia, Europe, Pontic Steppe, Russia, Levant region, North Africa, Swahili Coast, Australia, Sub Saharan Africa, Siberia, America (obviously) etc... And then very minor presence in Iraq, Iran, Mongolia, Manchuria, Kwarezm, Tibet (rapidly growing) etc... To fulfill the criteria on where Buddhism can spread, it should be noted that Buddhism to fare better than otl, it must maintain existing regions and then expand from there. Loss of major regions of power, could prove disastrous for the religion as a whole and further, political dimensions matter most...
In otl, Islam came into the world as a new narrative and socio-political system which uprooted the systems previously maintained by Rome, Iran, China, 'the steppe hordes,' Hindustan and China; in that, the Caliphate which declared in and of itself a revolutionary idea of expansion that shunned previous imperial and social concepts. Thus, it is no surprise that religions such as Buddhism which spread across Asia in a time of a fundamentally different narrative, would be shocked at the rise of such an entity. The milieu with which Buddhism spread is one of the Silk Road and interconnections made between Hindustan, a land of great population and varied luxury goods and a deep and long history of religious scholarship. This relative importance of Hindustan in terms of trade and exchanges, gave religions pouring outward from Hindustan more effective than otherwise and Buddhism capitalized extensively on this boon. Further, the religious atmosphere of deep scholarship and discussion that permeated Hindustan during the time of Buddhism's rise and subsequent periods prior to Islam, made the religion particularly 'fit' for the task of spreading outward.
However, this spread Buddhism achieved after its rise, was made by the virtue of both incoherent or unbalanced policy toward Buddhism among the states making up Asia (similar to Christianity and Rome, aside from the Diocletian tyranny against the Christian and Manichaen beliefs [among others]), often political support from rulers in favor of Buddhism, remaining importance of Hindustan and the status quo along the Silk Road network. Understanding these issues, we can assess Islam's effect on Buddhism and likewise two other religions that spread for similar reasons (Manichaenism and Nestorian Christianity/Church of the East).
-Islam in the sense of the first issue of incoherent policy, completely differed from the Chinese, Iranian and Hindu rulers preceding them. For instance, the concept of Buddhism's existence is considered to be idolatry for the worship of idols and promotion of a religion that is not protected. Thus, its existence once conquered is only at the benevolence of their captors. Further, Islam creates an eternal status within scripture, tradition and legal coding that prohibits Buddhists from holding any position within the state, from spreading their faith, from maintaining temples or monasteries or any public showing of faith. These restrictions are also not incumbent upon the state or Khilfah alone, but is incumbent upon individuals and non state collectives to administer the Shariah and rulings of Islam to those whom Allah has prescribed to be those of the disliked faction with whom you war with until submission is achieved. In relation to Buddhism, other than various massive structures (which the times in context, there existed no way to destroy such buildings without a massive amount of funds) such as the Bamyan Buddha or hidden structures within caves, all records that I know of in interactions between Islamic conquest and Buddhism, led to the desolation of various Buddhist temples and structures of faith. There are numerous examples of these happenings for Buddhist structures as well as other religions, such as Zoroastrian, traditional faiths, Hinduism, etc.... So, the milieu wherein Buddhism spread against varying jurisdictions with separate outlooks on how to combat and differing policy (over time, as in the Chinese Emperors differed in opinions on Buddhism; one would embrace the religion or take benign neglect toward it and another would actively seek its uprooting) toward their faith, waned or totally ended in the western spectrum.
-Regarding the second issue, the state support of Buddhism began to wane in the west rapidly as the Islamic armies defeated Buddhist states and peoples in war. Islamic armies under the Umayyad and Abbasid dealt major blows to the states professing or supporting Buddhism in Afghanistan, Northwest Hindustan and damaged the religious mosaic of the Silk Road and Central Asia. Blow against the Tang dynasty by both the Abbasid and Tibetans further hampered this status quo of the Asian continent.
-As a result of the Islamic conquests, the importance of Hindustan in the world market moving westward, declined sharply for a period of time. The reasons for this include the invasions that began to take hold on the perimeters of Hindustan in the 7th, 8th and 9th century then intensifying rapidly in the 10th century as a status quo of invasions of Hindustan for loot and power began. The nature of these conquests further was separate from that of the Hepthalite, Kushan, etc,,, in the fact that the previous entered within Hindustan as entities which molded with the Hindustani continent (Islam did but to a lesser extent and this varied by the Islamic polity). In contrast, the early stages of the conquests, Hindustan was increasingly separated from the western world in trade aside from loot captured and sent internally from within the Islamic world. In later stages, this became much more extensive as Islamic states conquered the majority of the northern section of Hindustan during the Ghurid and Mamluk period. Once the Islamic polities began to cover nearly the entire continent, during the Middle Delhi Sultanate phase, Hindustan's importance in trade surely began to raise and with it, Islam's presence among the Malay, Sumatrans, Cham and eventually across the entirety of the Southeast Asian islands and along the coasts of Burma, Thailand and Vietnam. However, Buddhism was too late for this development, as Buddhism likely waned in its strongholds within Hinudstan at some point in the 12th century or earlier. Regardless, Buddhism would not have gained in this development as it would be the religion of the 'dominated,' and despite the flourishing within Hindustan, Hinduism did not make any real gains outward as it was Islam which ruled the trade waves by the 12th century, especially in the crucial region of Bengal. In connection with this, the loss of any meaningful Silk Road, diminished with the rise of Islam in the region; even without the prior mentioned Islamic policy toward Buddhists, with the Silk Road breached in any way, the Buddhsist expansion westward is retarded significantly.
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After giving some reflection to why Buddhism declined so rapidly within Asia and westward in the face of Islam and a changing continent, we can discuss what can be done to increase the westward expansion of Buddhism and likely the overall expansion of Buddhism.
In Asia in the 7th century, we can look with our pencil (to manipulate growth of Buddhism) at the state of things. Assuming the Sassanid-Byzantine wars goes as planned, we see the victorious Byzantine empire while victorious on paper, gained a disastrous victory that presents itself more issues than benefits. Byzantium however, remains relatively powerful and without Islam do not see a collapse on their southern and eastern front as occurred otl, at least not so dramatic. To its south, Arabia remains relatively open as warring tribes vie for power or exist in confederation with one another to maintain trade relations. Yemen is further free recently from Aksumite rule and resumes its previous role as a middle man in the Red Sea trading zone with Hindustan and the Far East by proxies. Sassanid Empire of Iran is beset by the loss against Byzantium and in a precarious religious state, with great religious diversity and societal conflict. Russia and the Pontic Steppe is prepared for another stream of invasions from hordes in the East, especially the myriad of Turkic hordes. Khazar hordes already exist in strong numbers by the 8th century. Pechenegs follow shortly after the Khazar and the Slavic growth in the north coupled with the Viking expansions. western Europe as well, struggles after the collapse of Rome in the West and other ills, but also moves toward a more cohesive order than existed in the 6th century. Aksum itself is on the verge or eve of deep decline, with or without Islam as the land around it begins some time during the 8th century to experience prolonged famine and unknown societal pressures, some related to Islam and others to the nature of states.
Hindustan, the most important front perhaps, is a necessity for our purposes. Buddhism within Hindustan makes a fairly sizable proportion of the landmass, comparable to Islam in numbers, but certainly weaker than Islam in the political realm. For our goals, the preferred state which is possible, is Buddhism retains its status as a major faith in Hindustan, but never totally dominate. Dominance of Buddhism in Hindustan requires an earlier pod and one that does not particularly seem plausible regardless (few religions dominate an entire continent with such massive numbers of people,, China for isntance has neevr been dominated by one religion and Hinduism in India itself is a conglomeration of many different ancient religions and traditions molded over time). So, the goal is to maintain the Buddhist numbers in Hindustan at around 15% or so, with a large proportion of which having professions as merchants and nobility. With this, Buddhism can continue expanding with the natural assistance of Hindustani trade force. The results of this, is Buddhism will have an inevitably stronger pull on Central Asia, Indonesia, Southeast Asia, Muscat, Yemen, etc,,, than otl.
Afghanistan, very near the heart of Buddhism and the transition region between the three spheres of Central Asia, Iran and Hindustan, is the valve through land by which Buddhism spread. In the absence of Islam, Buddhism is already a powerful force, especially within the north of the region and in the city of Qabul. Buddhism's place here likely remains. However, the rest of Afghanistan is undetermined religiously, with some of the region practicing traditional faiths separate from the Dharmic and Iranic traditions such as in deep mountain villages, within the southern section of the region, the Zunbil traditional faith continues to resist to some degree Buddhism and other nearby faiths, Zoroastrian and other related traditions also predominate in various parts of the region. The extent to which Nestorian and Manichaen groups exist in Afghanistan is unknown to me. Their existence is certainly assured, though it would surprise me greatly that either predominate anywhere in the region. Zunbil in particular, seem to have a unique mixture of faiths with the central figures of the worship of a solar deity and presence of golden idols and great altars for sacrifice. Despite these basics on the Zunbil, their faith will likely remain a mystery to us. Thus, we will make some assumptions regarding it as a conglomeration of various nearby religions into some sort of traditional synthesis around a solar god and a strong tribal force to defend it. Buddhism holds some strong advantage in the region based on my assumptions; Buddhism holds strong advantages in terms of its trading traditions, relation with to its nearby east and a strong literary tradition. Whereas the Zoroastrian mosaic, represents either traditional groups in the region related to the nearby Iranian plateau or presence of the Sassanid force. In both instances, this may be an area that Buddhism can make inroads, by the decline of the Sassanid empire and Buddhist magnates expanding into these areas or trade force from Hindustan begins to outplace the Zoroastrian faiths present in relation to the Iranian ensemble. Holding the Manichaens and Nestorians out of the region is also done by limiting the extent to which Manichaens spread among the steppe hordes by an increased presence of Buddhism and then doubled with this, perhaps either Tang or Tibetan aggression toward Manichaen and Nestorian existences. In both these examples, Buddhism in the absence of Islam, holds massive advantages over these due to its presence in Hindustani trade networks. Buddhist expansion into the Zunbil is the more difficult. The Zunbils had likely already withstood Buddhist attempts at spreading and unlike the Zoroastrian groupings, resisted Buddhism without state support from Iran. Thus, it is plausible the Zunbil remain fro the forseen future as the ethnic and tribal faith of southern Afghanistan and Baluchistan, with mild presence of Buddhism. Traditional faiths separate from the previously mentioned groups remain with minimal Buddhist influence in remote locations and as syncreticized versions of other nearby by faiths.
As extensions northward, Central Asia immediately is another area of influence. Buddhism itself was already a force across the Silk Road by the time of Islam, but without Islam, while certainly benefits, will acquire more numerous foes. In otl, Buddhism competed greatly with Manichaeism in both Tibet and Central Asia. However, this was with a weakened Manichaen base as Manichaenism was utterly annihilated by the 11th century in the former Sassanid Empire by the Abbasid authorities. Despite this, my opinion would be that with strategic writings wherein the Tang remain ascendant in central Asia or conversely a Tibetan-Tocharian alliance to slow or limit Turkic powers int he region that could embrace Manichaeism or Nestorianism. Alternatively and for the purposes of our goals, Buddhism spreads through strategic reasons up to the author, to completely supplant Manichaeism and latch onto traditional Turkic belief or co-opt it and spread with the Turks westward not toward Iran, but across the steppes moving toward the Pontic Steppe region. This Buddhist movement westward is then coupled with tolerance by the Khazar elite and eventually spread toward the various parts of its empire now unhampered by Islam. Buddhism thus under the Khazar, gains a vital role in the realm and semi permanent placement. From here, the Buddhist faith could possibly spread northward into the various peoples there or still further westward into the Slavic and Viking realms. Perhaps to achieve this massive growth westward, Buddhism relies on incoherence or neglect on the part of Turkic, Slavic and Nordic factions to combat the faith and thus it makes various inroads as bringers of knowledge regarding both writings from the east and in other techniques. As well, the Buddhism in the regions moving further and further west and north, could find a place as powerful community ascetics and spread among local communities in ways not dissimilar to the Christian spread within Egypt. Coupled with this, is to present major issues in the Byzantine empire and preferably continued traditional faith adherence by the Bulgar Khaganate. One issue with any of this that I admit, it presents issues regarding the spread through trade Buddhism acutely gained historically from. Trade toward this direction is low in output, I would assume.
The Sassanid Empire is reeling from the loss, however, my opinion would be that the Sassanid power is not totally lost and a rebirth is possible or at least a temporary restoration. For instance, the view that the Sassanid fall to the Lakhmid confederates, I find to be extremely unlikely, much more unlikely than even the Islamic conquests of otl. The option for one seeking to end the Sassanid state as part of a tl, would be the Partian magnates revolting and capturing power. What this means is up to the author, but it could be a major point for the spread of Buddhism. Say, the historical Parthian tolerance of the past is returned, there could be a scenario wherein Buddhism could make a serious entry. One idea that I find possible, is for varying reasons related to just greater time and a degeneration of the stringent Zoroastrian/Iranian puritanism of the Sassanid, Buddhism makes inroads among the elite of Iran in a molding of the Iranian faiths with Buddhism. As a result, Buddhism makes an entry into Iran in a new form as a religion that can promote certain desired goals among the Iranian Empire and to assist in the difficult religious situation within the Empire. That being said, it is plausible too and more likely that Iran remains strongly Zoroastrian and supports loyal religions and opposes those that present a threat. This form of Iran would likely block the spread of Buddhism westward through the plateau.
As far as Arabia is concerned, it is important to separate Arabia into different subsections. To the southeast and east, the coastal east and Omani coast are each under Sassanid influence and part of that ensemble. Buddhist spread toward the coastal city of Muscat and other smaller ports such as Hormuz or along the eastern coast, likely occurred to a small degree otl. These areas in otl, became hotbeds for religious dissent and fanaticism during the Islamic period and thus the religious makeup of these areas could be ripe for the entry of Buddhism to be transformed into certain syncretic and fanatical subsects as is common within Buddhism. In Yemen, there is a trinity of religious adherence. Unlike the east, Yemen displays less in regards to the religious dissent as seen otl and in this tl, will have less chance of acquiring the same traits as that region. Yemen as a whole was predominately traditional faith, with powerful Jewish presence and limited Miaphysite presence. After the fall of Aksumite power in continental Arabia in the late 6th century, Yemen relapsed into a primarily traditional faith rule across its land, which previously was held by the intensely Jewish Himyari lords (decimated in wars with Aksum). The ability for Buddhism to spread certainly does exist, with enough well written reasons and power of Buddhist merchants in the Indian Ocean toward the Red Sea. Then there is the Nejd and Hijaz, each presenting different problems. In both cases, my opinion would be that Buddhism would have very low chances of maintained expansion here without extreme growth in other realms. My opinion would be that these regions stay predominantly traditional Arab, with the moderate presence of Judaism and Arianism within various areas of the Hijaz north (Christianity was near absent from Makkah).
Regarding Africa, Buddhism making gains in Yemen would give the impetus and valve with which Buddhism spreads toward what became the Swahili coast. However, the extent to which Buddhism can dominate in Yemen, I find mostly unlikely, yet possible. Aksum's decline also presents a possibility, albeit very low. Yemeni and Arab expansion into Ethiopia and Somalia likely occurs to a much greater extent than otl. If Buddhism can latch itself to some of these Arab conquerors, the religion can implant itself in various parts of Ethiopia and coastal Somalia. It should be noted, the Arab conquerors, I do not suspect will convert to the religion of Ethiopia, the Christian faith in Ethiopia, I do not estimate as being a manifestly powerful entity and will be unfavored by the Arabs who have known relation to the religion and can combat it. A traditional faith and Jewish faith revival thus could occur in Ethiopia as Arab conquerors put stress upon the Christian authorities of the region and slave raiding damages the numbers of Christian adherents.
In conclusion, this would be some of what I could imagine and see in terms of a fairly well empowered Buddhism. Any feedback is appreciated.