Specifically, did they maintain their claim even after Nicaea took Constantinople and became widely recognized as the Byzantine Empire?
While Trebizond may have initially claimed to be the legitimate successor state of the Roman Empire during its inception, using the ruling Komnenos Dynasty's Imperial heritage as further justification, these ambitions gradually faded over time.
Trebizond initially wrestled with the Empire of Nicaea for control over Paphlagoinia in the years following 1204, but were eventually defeated and retreated to their core territory. By the time the Empire of Nicaea was able to reclaim Macedonia, it was pretty clear who was the dominant Roman state. However the Komnenoi of Trebizond still held on to their Imperial claims.
Following Michael VIII Palaiologos' recapture of Constantinople in 1261 however, things changed. Michael felt pretty emboldened by his suprising recapture of the city, and sought to legitimize Nicaea as the one and only, legitimate Roman state. Michael sent envoys to Trebizond and asked them to renounce their claim to the Roman Empire. In exchange Michael would offer one of his daughters in marriage, and bestow the Trapezuntine Emperor with the title of Despot.
Emperor John II, suprisingly agreed to the proposal. After marrying Eudokia Palaiologina in Constantinople, he formally changed his title.
From this point onward the Emperor of Trebizond was no longer;
Emperor and Autocrat of the Romans
but rather;
Emperor and Autocrat of all the East, of the Iberians and of Perateia
As such Trebizond recognized Nicaa as the sole, legitimate Roman state, and renounced their claims entirely. Despite outliving the Empire itself by nearly a decade, they no longer had a claim to the Empire as a whole.