What is the actual origin of the prefix 'bi-'?

Because we can trace the origin of English bi- to Latin bis "twice", and from there back to Proto-Indo-European; and we know many cognate Indo-European words, such as Latin duo and English two. In Proto-Italic, bis was still *dwis. The change of dw into b happened later, forming Latin bis.

I don't know anything about Basque except that it is a language isolate, possibly part of some larger pre-Indo-European linguistic family, but I would rather expect Basque to have borrowed this word from superstrate Indo-European languages (Celtic? Latin?) than the other way around.

Compare Latin bellum "war" (whence English belligerent), from an older root *dwe-, which can still be seen in the alternative form duellum, whence English duel.

From De Vaan's etymological dictionary of Latin: