Unfortunately I don't know all that much about the nobility of Provence, although I presume King Boso had significant local support to be able to claim the kingship in the manner he did.
Strangely enough, not that much : contrary to the possible (but disputed) royal claims of the dukes of Gascony that would have been buiilt both on an established clientele and with ties to the Kingdom of Pampelune, Boson's regnum is largely opportunistic, and non-descript.
Boson was initially more of a viceroy of sort in Aquitaine, then Italy and critically Provence (where he gain some honores) for Charles the Bald : his power came from his proximity with the imperial court than his honores (which he inherited as he didn't kept his former Aquitain's). He timely leaved the imperial boat at Quiery in spite being called by Charles, and Boson joined the general nobiliar revolt in Aquitaine.
It might be strange for an "imperial man" but he was not really hesistating to join up with local potentes if he had to, especially in a really
fin de règne ambient.
I'll spare you the details of conflicts about who recieved which honores, and how everyone felt screwed by Louis II, but it eventually turns out that nobody really wanted to proclaim Louis as emperor in Arles as it was initially planned, while he became close enough from Carolingian circles (on which he was tied matrimonially) Boson not really making huge efforts in this direction and more focused on his own interests before a mostly benevolent (for his person) pope. While Guilhelmid collect honores (mostly taken from Adalelmids,, convenient scapegoats), Boson is trying to be crowned king of Italy which fizzles almost immediatly.
The shaky succession of Louis II, between Louis and Carloman fron one hand, Louis the Younger from another, allows Boson to make another attempt at kingship successfully this time.
He's supported by local nobles, but more significantly by the episcopalian community including outside Provence. Most of his nobiliar support probably comes from Charles' clientele, altough Boson barely manage to have half of Charles' rengum, the half more devasted by the troubles of the late IXth (including, notable, Saracenic piracy).
Giving that he joined back with Bernard II Adalelmid, that kept his honores in Provence against Guilhelmids, we might have a rag-tag support from bishops wanting some stablility (especially from someone having some pontifical trust), and a nobility that may feel is not enough taken in consideration.
Note that the royal title of Boson is non-descript : it's hard to say if he was looking at an italian (the pontifical favour turns after his royal election, so that's more or less gone) or aquitain kingship, however, but in the misdt of troubled and quick succession, Boson manage to carve out an opportunist and vague kingship whom only the divisions, undecision and early deaths of various Capetians really save; and not even completly (loosing most of formerly Adalelmid-held Burgundy as well as Lyonnais).
It's a mix of opportunism, succession crisis, and networking, more than just local support. Giving the historical consequences for the late IXth and Xth century, it's quite amazing. (Boson really makes me think he was sort of Carolingian Charles d'Anjou).
After reading what you've posted and poking around the Aquitanian genealogy (which I knew little about), I certainly agree that a Guilhemid survival, particularly in the form of a son by Engelberga the Bosonid, presents a reasonable chance of extending Aquitainian influence or even rule there. It's too late to avert the usurpation of Hugh, but Hugh's demise would leave only Charles-Constantine and his descendants (and his supposed in-laws at Vienne) as serious local competitors, and my impression (not based on much, admittedly) is that their rise in the later guise of Counts of Provence owed much to the breakdown of royal power after Hugh's Italian venture and subsequent death and its replacement by the feebler and more distant royal presence of the Burgundian Welfs.
Indeed, Aquitain genealogies tend to be blurry at best (the Raimondine traditional genealogy at least left out two Raimonds and one Pons in the list, and they are one of the most known families), but Boson Guilhelmid is quite an interesting PoD.
I was less expecting to butterfly away Hugues's presence, than having Guilhelmids appearing as Carolingian-Bosonid "loyalists", and push Hugues out of Provence as Rodolphe II did IOTL, appearing as supporters of Louis III. Their ambitions are unknown, but, as said, their relatively affirmed territorial strength doesn't really makes me think they would go for the royal title, and just takeover Hugues' honores in Marca Vienensis.
Now, would they try to pull the same policy Bernard II Guilhelmid did with Carloman II (pulling a regency for Louis III, then Charles-Constantin) or try a bid the for throne is let to imagination : but geopolitically, I'd say the former makes more sense (and would be more helpful ITTL), altough Boson Guilhelmid could possibly not realize how much is position in Aquitaine is still dependent from a relative delicate balance (especially if Ebles Manzar still is Count of Poitiou ITTL, which might be better to care of, as pointed in the post 28).
That said, I'm not too sure how Welfs would react to this, in a TL where Robertians are weakened, and Burgundian Bosonid not as much able to benefit from their support.
Your other alternative, a Bosonid rise at the expense of the Robertians, has its attractions, although it seems to me that a Bosonid on the throne may be only marginally more successful at maintaining the profile of the Carolingians than the early Capetians were (which is to say, hardly at all). One wonders whether it might be easier to possess the imperial title first, before the Francian kingship, so as to present oneself as "Carolingian" in title if not in lineage, but I'm not very well acquainted with the manner in which the pre-Capetian French chose their kings.
Yeah, I was afraid of this, altough I saw less a Bosonid rise at the expense of Robertians, than Bosonid inheriting Robertian networks and part of honores. Not unlike Arnulfids did with Peppinids in the VIIth century.
Now for the choice of Western Francian kings...It would be relatively agreeable to say it was elective, but it was a really unformal election, which several potentes didn't acknowledged (Aquitain potentes tended to refuse to acknowledge Robertian kings) : it eventually depends a lot on what happened in northern France, closer to royal circles, at this point, the succession being eventually more or less settled (sometimes not, as with Charles III acknowledged in the southern part as king of Franks, Aquitains and Goths) with peripherical potentes in exchange of honores, beneficii or alliances; all of this with back-and-return for legitimisation to German kings : a wonderful, chaotic mess.
So basically, it's maybe less of an election, than an appointed kingship trough several potentes, usually close to royal networks or powerful enough; and usually a contested choice par Aquitains, Goths, Champenois or Bourguignon, except with tractations, if it involved another dynasty than Carolingians (a very, very accident-prone dynasty : as you said, candidates were running low, so it would be nice if we had more of these).
The distinction to make, I think, is that while Alberic was quite happy to "reform" the monasteries into something more creditable and useful to him, the career of Odo in Italy made no immediate impression on the Papacy itself, which under Alberic was actually quite creditable morally but quite restrained politically.
I agree, but the greater autonomisation of the clergy, monastic or episcopalian, would still have a strong drive ITTL. With time, especially with the hot seat that was italian kingship, the idea of a more clean clergy (I entierly agree that the Tusculani Papacy really had a bad rap on this regard), and less devoted to great families when it come to secular matter is going to be particularily important and pervasive within Christian society, as IOTL. Not just within the clergy, but among the nobility as well, where you did have an echo for re-evangelisation of feudal society historically.
Hence one of the reasons why I think that a later Western French intervention in Italy might be preferrable to form an HRE-equivalent.
At the very least the emperors will have to keep reliable men in Tuscany and Spoleto to keep them from "pulling an Alberic" (that is, Alberic I, not II) and using their power to assert themselves in the city.
Is there no chance to have Tusculani simply wearing out, in lack of royal legitimacy to strengthen their principalty? After all, the number of families in the late Carolingian world that could have had a go for territorial survival, but crumbled because of the lack of legitimisation for their extension is quite significant. Or is it too late to really play any role in Italy?
How do you see a Cluniac reform being "stronger" under royal/imperial Guilhemid patronage? Just in terms of the resources they are granted by an ascendant Aquitainian state, or in some other way?
Well, partially so : after all Cluny foundation is both due to an initial re-evangelisation of French society, and because it allowed Guilhèm to really strengthen his presence in the part of Burgundy he held. I could see something similar, within a Provence where Guilhelmids represent a strong influence, more regional foundations, earlier.
Similarily (copy-pasting post 25, if you allow me), it's worth noting that the moral decline of the clergy and its "nobiliarisation" in France went in par with the decline of the Carolingian dynasty, Robertians hardly behaving differently from other nobles until Robert II.
ITTL, Carolingians could end with a stronger clergy that what existed IOTL (especially without Ottonian influence on Xth century clergy), patrially due to a concious and maintained policy as the episcopalian support was a strong feature of their own legitimization in France. On the other hand, this clergy isn't as dependent on royal (or there imperial) power that it tended to be in Germany : the regalian role of the clergy is more present in the east of the kingdom (and probably Lotharingia as part of F-HRE ITTL), and tends to disappear quickly more you go west and south.
It's hard to say if the regional councils would be maintained as IOTL, altough I do think that strong potentes as Aquitaine would. But the Peace of God movements would be likely changed due to a possible (but really relative) imperial appeasement : it wouldn't be butterflied away, tough, as a F-HRE would be still particularily troubled in several regions, with an emperor not rally able to hold out conflicts in peripherical regions if at all (while still more able to do so that the Xth century IOTL).
We might see a dynamic where a stronger Cluniac movement benefits Carolingian dynasty, which in turns legitimate Cluniac influence in Italy, etc. A bit far-fetched, I agree as on the other hand, it would mean create an imperial-pontifical relationship out of it, rather than strengthening reforms because of a previously existing imperial-pontifical relationship. Hence why I'd prefer your opinion on this as well, as it depends a lot on Roman policies : what you said tough, doesn't really makes me confident.
The Lombard duchies in the north didn't really survive the Carolingian conquest (unlike those in the south, which remained extant until the Norman conquest), and as a result the territories of the proceres/magnates of Italy in this period are very often either ad-hoc distributions of the formerly royal territory or myriad assemblages of estates and rectorships (such as what the Hucboldings acquired in Emilia/Modena). See, for instance, the division of south-western Lombardy which developed under Hugh and Berengar II (apologies for the poor picture):
Any reason why the Carolingian conquest was this destructuring in Italy? Usually, they kept most of the frames in took over regions. In fact, why did it happened in most of Carolingian Italy except Spoleto and Tuscany (I would have said Friul as well, which seem to have been distinct already before the Frankish conquest, but...).
I'm not exactly sure to understand how these political formations did legitimized themselves without at least a token power : or was the chaotic kingship the expression of this need of formalisation of honores which without inner jnbalance, had to be short-lived?
The late 10th century is not too late for a foreign king, Guilhemid, Bosonid, or otherwise, to come in and remake the map. Tuscany may remain Tuscany, but everything else is probably fair game to be carved up and apportioned among the king's supporters, which is exactly how things went down in southwestern Lombardy c. 950 as mentioned. You can't get all that much more "disintegrated" than OTL aside from breaking up Tuscany as well.
So, we might need some various non-Carolingian kings (but not emperors) messing just enough with Italy until the late Xth/early XIth, and allowing a Carolingian firm takeover?