I don't know the degree to which Central Asian trade routes figured into the Roman economy, but the disruption of overland links to China are very apt to be at least partially offset.
The Black Sea trade in general should be developing rapidly, and the short Volga-Don portage means even the Caspian can get in on that. For the Romans, that mostly means an ever-greater supply (at cheaper prices) of primary products. The fact that Black Sea rivers make high-volume transport very practical even for bulk goods, and that the Roman economy is highly maritime pushes that even further. Given that the Roman economy is a highly urban and mercantile one, and its geographic position, this trade can only be accelerating the growth of artisans, commerce, and early manufacturing (such as it is).
More than that, the Ottomans lost their foundries and supplies, and the Georgians also need to rebuild their forces as well as Baku. The most practical option for much of this comes from Roman production, which must be growing in capacity all the time. Fortunately, the Basra-India-SEAsia route seems undisturbed, so Antioch should still be getting plenty of traffic (and the Ottomans can pay for Roman imports to recover).
For the moment, I don't think money is a pressing concern for the Romans, especially not compared to manpower. And if there's one thing that Renaissance history has taught me, it's that money can definitely buy an army--or at least the units for one. Perhaps the Russians and Georgians might allow some recruitment if Constantinople bears the cost.
Further, others have mentioned Vlachia and Hungary either intervening or simply taking advantage... But it's worth noting that military intervention isn't the only way other powers can get involved. The Avignon Papacy can wield a lot of influence, and both it and Genoa could move quite a bit of money if the Romans seek a loan. More than that, Venice needs to be very careful not to give Aragon, Sicily, Genoa, etc a casus belli by damaging or disrupting their merchant quarters or looting their property--The Roman fleet might be caught by surprise, but Venice almost certainly doesn't have the numbers to overawe Constantinople's navy, support armies in Italy and the Aegean, and defend against western intervention all at the same time.
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Looking back at this post, I suspect Emperor Theodoros and I would get along splendidly, and not a single coin would escape.