Sure a fawn can survive, so could a 4 year old child without his or her mother. But I guarantee if you kill mama with a fawn, that fawn is way more likely to die, either from natural causes or from getting shot.
This may sound brutal, but I tend to shoot the fawns/yearlings and let mamma walk.. I like having mature hot does running around during the rut. Sorry if that offends anyone, but it is my logic and I don't have a bambi complex..