Česky bych spíš řekla Narodil se se třema rukama. Tohle se i špatně vyslovuje. I would more likely to say Narodil se se třema rukama. I think this sentence is bad for pronunce.
Can I use the passive construction here: ''Byl narozen s třema rukama'' ?
No.
Actually it is correct as well
Gramatically it is possible but it is not very natural.