Heh, if we are looking for examples of Latins using Romania-one need not go much further than Imperium Romaniae (
Latin Empire post 1204 itself) itself. Sure, made up of people literally on those lands, but a bit telling nonetheless. And before you say this was politics to keep the local population on the side-you have waged this ridiculously brutal war against these people, imposed the Latin Church on them, and your top concern is to call yourself Roman so that people don't rebel/defect to the Orthodox successor states? Doesn't quite seem plausible that the demonym mattered over religion-especially since Orthodoxy was one of the pillars of being Roman by this period.
Calling it Romania or not was likely always a matter of convenience. The Latin Emperor of course wanted it as it elevated his position to something one could believe was comparable to the HR Emperor-which Emperor of Constantinople would not quite do. The heretical enemy on the other hand did not quite deserve the name etc etc.
The Persia/Iran situation is interesting, but I am not sure if it is quite comparable-since Persians and Iranians were never viewed as distinct in the western psyche, while people seem to have a rather sharp dividing line between Greeks and Romans ( I blame pop history lessons but w/e). The circumstances however matter quite a bit, if we have a situation with the Emperor remaining the preeminent Mediterranean power of Christendom, there would likely be no 4th Crusade and subsequent rise of "Greek nationalism". Consequently, Greek would likely be viewed as an insult, versus the case OTL-when it was slowly being accepted as a substitute (as we can see from Theodore II Laskaris etc). So the latter case could lead to a situation like Germany/Japan/China where the demonym is distinct from the commonly used name elsewhere.
On the other hand, a surviving powerful Empire which dislikes being called Greek could probably normalize 'Romanian/Rhomanion' to an extent. Especially if they get into a Franco-Ottoman style relationship with some European powers (perhaps allies of Alt-Protestants against Papal supremacy/Catholics) who will start calling them Romans or a variant in their own self interest, eventually normalizing the term. And if one of the allies build a global Empire with the same reach as Albion OTL, then the job is fairly done.