Really few chances for that to happen.
A really important part of the Roman Emperor legitimacy, even when it comes to Barbarians, was that it would be Roman.
Giving that roman citizenship is distinct (and while not wholly opposite, distinct enough for that gaining one make you abandoning the other) from Barbarian, and that imperium was definitely associated with Roman citizenship, you do have a huge problem that is not about ethnicity itself, but political identity. (Barbarian "identity", and a fortiori kingship, was seen as opposite as Roman citizenship, if complementary in the roman roganisation of the world).
Even massively turning to Nicean Christianism (rather the converting, due to the closeness of beliefs with Homeism) wouldn't really help : if something it could make them more obedient to the Roman Empire in Constantinople.
I admit that it's pushing one possibility, but the conversion of Frankish kings didn't really provoked their "official" romanisation (while hastening the fusion of population towards a post-imperial romanity), at the contrary (Gregorius of Tours points that's less and less Romans in Gaul) : and that even if Homeism was a key part of Gothic identity up to the VIIth century (which means that a large switch would provoke issues, as can be pointed into Visigothic kings religious policies in the VIIth)
As for "Holy Roman Empire", the whole concept probably gets butterflied away with a Gothic hegemony : it was mostly the result of Post-Carolingian times, with Charlemagne never proclaimed himself "Roman Emperor", but "Emperor ruling over the Roman Empire", a really different concept, not claiming romanity or being roman, but the imperium over Christians, at the likeness of what claimed Constantinople.