Additionally to their Germanic culture/language (which is not much to say, as romanized as the Goths were), their biggest identitarian characteristic was their christian arianic faith. If they decide to style themselves emperors, they could use that as a mark of prestige and opposition to the catholic eastern empire. Of course, as IOTL Visigothic Hispania, that would alienate them from the roman popular base, unless...
-they are more successful than IOTL about attracting peasants and villae lords to themselves.
-they pull a Reccared like IOTL and convert.
So, we could have ourselves a Gothic Arian Church set up over a revived Roman Empire in the west, with a Germanic aristocracy?
Maybe the Arian church could start sending missionaries and converting Catholics in the West, combined with less tolerance for Catholicism, under a longer-lived Theoderic the Great, or a son that (as far as I know) never existed. This sounds like it has all sorts of places it could go.
Early crusades, a deposed Bishop of Rome/anti-popes, religious wars back and forth between the Ravenna and Constantinople.
And, of course, Theoderic IOTL allied himself with other Germanic states, via marriage or threat of invasion, so...
Visigothic+Ostrogothic Kingdoms+Frankish Kingdoms+Vandal Africa = Gothic-Roman Empire?
Or at least...
Visigothic+Ostrogothic Kingdoms= Gothic-Roman Empire, with Frankish and Vandal Kingoms as allies or even client states.