As Fabius Maximus pointed out Ulfilas converted the Goths.
The real question is "Why doesn't they convert to Arianism?".
Do they remain pagan or do they convert to another brand of Christianity?
Considering that by the time Ulfila started preaching the other major heresies of Christianity didn't exist yet, there's not a lot of possibilities.
I think they would be assimilated faster, gaining a lot more support than IOTL in particular the Visigoths, but also the Ostrogoths.
A nice butterfly could be Eutharic living longer thus starting a unification of the Goths.
Probably the Western and the Eastern Church would break up sooner.
With these premises I don't see any successful attempt of reconquest from the East.
Honestly, the best part of the Goths maybe becoming Catholic is the fact that a proper Latin script for their language may had been developed, instead of this weird ass Greek-Rune hybrid I have to deal with...
It can get worse. If a Christian missionary to convert the Goths before Ulfilas perhaps crafts a only rune-based script then we might see a Church Gothic Language substituting Latin and Greek in some areas as happened with Church Slavonic Language, but I wouldn't bet on this.