The ERE of this period was politically stagnant and utterly demoralized. Throwing money into the treasury is not going to fix an ancient, crumbling ideological structure. Political and religious defeats only pushed towards a more defensive, conservative political outlook.
The ERE needed new blood and new ideas. The early Ottoman state had everything Byzantium lacked to be a great power on par with any in the West. To co-opt that dynamism would be to begin to reform the system that would lead to recovery.
If it was an "ancient, crumbling ideological structure", I would agree.
Since that's not why the ERE fell, I'm going to repeat: Without money, no army and navy. Without an effective army and navy, anyone like the Ottomans can come along and tear off pieces of what remains of the Empire's territories in Anatolia, anyone like the Catalans can run amok when not paid, Venice and Genoa can demand anything and get it because there's nothing the Empire can do about them.
No matter how inspired or how visionary or how dynamic the
basileus or his generals are.
Take Andronicus II, since he's the Emperor at the time of the POD. Why did he cut the army to the bone and just about eliminate the navy? Money.
Restore the Empire's finances and such a move wouldn't be dreamed of.
From Timothy E. Gregory's
A History of Byzantium
"Michael VIII died in 1282 in what appeared to be very good condition. To be sure, Byzantium had re-emerged onto the stage as a major player in international affairs. Nonetheless, his successors were completely unable to maintain the political and military power of Michael's empire and it is an open question to what degree his policies were responsible for this decline. On the one hand, Michael had expended enormous energy to restore Byzantium to a position of power, and this had possibly weakened the broader fabric of the Byzantine economy and state. On the other hand, we must be careful when we blame the successful Michael VIII for failures that ook place under the rule of his successors. Ostrogorsky is clear in his assessment of the situation: "In reality there were more deep-seated reasons to account for the rapid decline of Byzantine power . . . the internal weaknesses of the state were incurable and increasingly internal pressure drove Byzantium irretrievably toward catastrophe" (p 479). In Ostrogorsky's view, the rise of the Ottomans and Serbia took place at a time when the state had been weakened by the expenditure of Michael VIII, and he notes that it is 'these momentous factors in foreign and domestic politics, and not the personal qualities of its rulers, which really account for the decline of Byzantium" (ibid.)"
A state not able to take the offense is not likely to be thinking offensively.
Meanwhile the predators (the Ottomans OTL being the best example) can feed on what remains of its resources and as they grow stronger, the ERE grows weaker.
But that doesn't change by having what was the Ottoman state co-opted into the ERE.