Even if Byzantium succeeds in regaining control over territories lost further to Manzikert, the situation in 1260 is in a critical point in West. The feudal system is over its end, beginning the systemic crisis that we called " Crisis of the late Middle Ages ".
Not being a real specialist of the economic questions I would have difficulty in developing it in detail.
But simply put in 1260 the situation is the following one: in the campaign the farmers are too numerous and the exploitations too small for allow an increase of the production.
Consequently while arrive the " bad years " (from 1301) the production collapses because of the bad harvests generated by the bad climatic conditions. What leads to a rise in prices of raw materials as well as fall in the poverty of numerous small developers.
This rise in prices has consequences on the whole economy, pulling a contraction of the exchanges and a recession. This coupled with a phenomenon of hoarding of the currency in which is engaged the powerful, leads to a deflation of the value of the currency what increases the recession and pulls the phenomena which we know today well: unemployment of mass, political and social instability etc. etc.
Interestingly, this could end up resolving some of the Byzantines problems - and I'll explain why.
Assuming that the Byzantines can resolve their administrative problems - through a larger bureaucracy, more paperwork, [insert appropriate solution here] - this mass-unemployment could cause an interesting circumstance.
The Byzantines with their relative wealth, and less populated countrysides, at least from my understanding - could offer lands to all peasants who migrate to the country - I mean, this is ambitious, and difficult, but if the Kings of Europe recognize that they can't handle their populations and the Neo-Byzantine Empire asks, then they could use this swarm of peasants to repopulate the countryside, and act as a major recruiting ground for their forces AND reduce the taxes on the peasantry in general - then it helps solves problems for both the Romans and the Latins, and repopulates those areas whose populations are pretty low. The biggest difficulty would be language, but that isn't insurmountable if there is a focus on bureaucratic reform already - training some bureaucrats to speak a
lingua romana could be useful if the immigrants aren't learning Greek.
Oddly enough, the Roman Empire reduces the issues its trade partners have, boost their own manpower base, and recover from their previous history much more quickly. Now the influx of Catholics is a potential issue here, but I don't know if it would be easy to convince those immigrants to convert, but even if they don't and the state restricts persecution, then that is a lot of problems solved.