My understanding is that one of the chronic problems of the late Byzantine Period was its ability to raise an army. However, it is also one of those states that had quite an advanced economy for the period, akin to Italy in some respects, and deeply involved in producing and trading manufactured goods.
This suggests to me that there could (arguably) be a labour shortage as well as a manpower shortage, especially in the wake of the Black Plague.
Could the Late Byzantine Empire have somehow portrayed itself as a land of opportunity - essentially guaranteeing rewards to those who immigrated to the Empire - the plan being to recruit these migrants as members of the Roman Army, and effectively conquer the land to both restore the Empires fortunes, and pay these people with plots of land.
This could apply to individuals, minor lords and their retinues, etc. Effectively part USA part Christian Ghazi.
Admittedly we're going to see Catholic-Orthodox issues, applying pressure to the Unionism conflict in play at the time, but say we do this during the early years of John V Palaiologos, could we see this strengthen the Byzantines enough to see a reversal of their fortunes - even if it meant being a Catholic Byzantine Empire?