For scenario #1, I think it is hard, but not impossible. It is only a matter of few things going somewhat differently for both Byzantium and the Ottomans, which is far from impossible. In OTL, everything seemed to go right for the Ottomans from the founding of their state until Tamerlane's invasion, and even then, after the destruction caused by Tamerlane had ceased, the Ottoman history between about 1430s and until about 1600 can be only described as an Ottoman-wank.
Presuming that in order for the Ottomans as we know them to emerge, the POD has to be after Osman's birth (1258, if I recall correctly), an interesting possibility in the Nicaean Empire pretty much presents itself. 1258 was also the year of Theodore II Lascaris' death, which happened in August - meaning that giving Theodore much longer life may not rule out Osman from being born, ascending to power, and being more or less the same person that he was in OTL.
Now, there is a reason Theodore is important. Unlike the Palaiologoi, the Lascaride policies towards the aristocracy were much less lenient, and actually encouraged the creation of a moderately powerful class of small landholders, some of whom were called on to defend the Empire from the outside enemies as the akritai - an institution very similar with the Turkish ghazis of the time. In OTL, death of Theodore (who was about 36 at the time) led to the rise of Michael Palaiologos, who usurped the throne as Michael VIII from Theodore's young son John IV, and founded a dynasty that lasted until the end of the Empire. Given Michael's pro-aristocracy policies, and him allowing the Turks to get strong in order to keep his political enemies more or less divided, a ruler who favored the middle class instead may leave the Empire's eastern frontier much better defended.
This means that providing Osman still rises to power (which he technically should - I doubt different Byzantine Emperor on the throne will change Osman's character very much, especially since the recovery of Constantinople is still the #1 priority for the Empire for the immediate future, and consolidation of territories is not close behind), he will not see the western Anatolia as easy pickings. Instead, ALT-Osman will probably look to the east, as it will simply make more sense.
In a meanwhile, let's presume the recovery of Constantinople is delayed by a few years, if only because the Lascarides are somewhat more careful, and don't have as much of a need to pull a publicity stunt as the usurper Michael VIII did in OTL. This also gives Theodore II more time to consolidate his holdings, and to ensure that when his son John IV does inherit, he is old enough to rule in his own stead. Given that John was born around 1250, let's have Theodore live until 1270 or so, thus giving Theodore 12 more years. This would not be too unreasonable, given that Theodore's premature death was due to his epileptic condition, and with the condition of medicine at the time, he would probably not live a very long life... but if he even manages to make it to 50 or close enough, it could be enough to drastically change the fortunes of the Empire.
So, let's say Constantinople is recovered around 1265, and Theodore spends the next five years consolidating his holdings, and managing to spread the institution of the akritai or similar throughout the Empire, while breaking the power of the aristocracy in a meanwhile. When around 1270 John IV inherits the Empire, he is around 20 years old, and his position on the throne is secure.
Now, John has to contend with the rivals to the west - the Bulgars, the Serbs, and the remains of the Latin ambitions towards Constantinople. As the Empire under Theodore II seemed to be more than capable of dealing with the Bulgars, I see little reason why the Empire under ALT-John IV would not be able to do the same. With the akritai more or less holding the eastern border secure, John IV can at least manage to keep a semblance of order in Thrace and in Greece.
In a meanwhile, by 1280 or so Osman starts building an empire amongst the Turkish tribes of Anatolia. Several attempts at incursions into the Imperial territory are met with heavy resistance, and eventually Osman decides that it is not worth the trouble, with strong Emperor in Constantinople, and with reasonably well-defended frontier. By 1300 Osman carves a sizeable kingdom for himself and his heirs; by 1330s, when Osman himself is gone, his successors are reasonably capable, but not to the level of OTL Osmanli (who had an incredible number of excellent leaders one after another) - remember, until Alaeddin's military reforms under Orhan I, the Ottoman military ascendancy was far from assured. Here, the Ottomans do not manage such reforms until much later in their history, by which time it is more of a standard practice.
So, what we have here by mid-XIVth century is a stronger Byzantine Empire that can hold its own, even if not truly expand... a weaker, but still powerful Ottoman state... both in a relative stalemate. I can imagine the Byzantine ambitions in the Balkans, considering the constant threat to their realms from the Bulgarians and the Serbs. In a meanwhile, the Ottoman ambitions would lie further to the east, as ATL-Ottomans are weaker than OTL without Alaeddin's military reforms, and with different set of leaders, some of whom might be brilliant, but not to the level of OTL Ottoman rulers of the era.
Also, let us presume that the Lascarid dynasty either continues uninterrupted for several centuries without major disasters (I know, it requires a lot of luck, but then again, chaos theory may result in no major, Empire-breaking disasters happening here - since this is the goal of this challenge, we may as well presume that), or gives way to another dynasty that more or less continues their policies, with only a small period of civil strife in between without having to extensively invite foreign involvement. The situation in the Balkans may reach a stalemate - I don't see this weakened Byzantium (although stronger than OTL) taking all of the Balkans (they simply do not have the power to do so without breaking the Empire economically and leaving it open to an incursion from the East), but they can definitely hold their own and keep the Serbs and the Bulgars in check. In the East, several provinces and border towns keep on changing hands, but for the most part, the situation is stable.
The ATL-Ottomans do not conquer as much as they have in OTL, but at the very least they match the old Sultanate of Rum. When Tamerlane or an equivalent shows up around 1400 (and that is a big "if", but let's presume there is someone very similar to him arising about the same time), the Ottomans are knocked back a bit, but not thoroughly destroyed; the Byzantines suffer a few military defeats, but eventually pay ATL-Tamerlane off, and since he dies shortly thereafter (again, a lucky break, but then it is not unheard of), do not suffer permanent damage. Perhaps the Byzantines, being that they are less affected by this invasion, manage to gain some small advantage over the Ottomans, but that advantage is soon reversed as the Ottomans gain strength again.
Now, the only thing remaining is to leave this relative stalemate more or less unchanged for two hundred or so years. The trick is not to let either the Ottomans or the Byzantines to get much more powerful than the other, or not to get any third major power involved that drastically changes the overall power situation. Again, with a lucky set of breaks it may be possible.
So, does this work?