I think Roemig was trying to pretend the GTVA didn't know about the defection of the Trinity and therefore decided to play the "I did this on my own, we stumbled into this portal and warped out without telling anybody" card. Eventually, the GTVA did find out about its (more or less) formal defection to the NTF and we all know what happened afterwards.
So yeah, it doesn't make a lot of sense for the Trinity to display the NTC prefix in its nameplate but we're talking about a change that affected just one letter so the crew probably had the time and resources to do it themselves in the ten or so days between the defection and their expedition into the nebula system.
I don't know whether the FSU just implemented Nyctaeus' NTF reskins - those are a whole different matter. It does make sense to have proper NTF skins for the regular fleet, but the circumstances of the Trinity's defection are such that the NTF skin should not be used for this one ship, IMO.