Joseph: That is incredibly unlikely as none of the Roman sources or ancient historians even reference, in any way, his conversion.
I find it most likely that, if you want to reconcile the Gemara’s account with the history that we can tell, he just ran off and disappeared after being condemned for being a traitor to Rome. Everyone assumed he killed himself (I searched but I don’t think anyone mentions a funeral). In reality he became Jewish and lived in anonymity somewhere.