I haven't had a 3 legged basset, but I have had a 3 legged lab mix. He lost a front leg and recovered just fine. He was still able to run, catch frisbees, and even swim a bit.
In the case of a basset, I think it would be easier if he's of lighter build. Also, a rear leg would probably be easier to deal with than a front. If your vet thinks he can manage okay I say go for it.
My vet recently told me of an elderly basset who had a pathological fracture of the femur and because the bone just shattered it wasn't repairable. However because of her size and weight they didn't feel she would be able to walk on three legs following an amputation and subsquently was euthanized.
I think a lighter boned basset would have an easier time adjusting to 3 legs.
Sorry you are having to face such a difficult decision.