In a simple elastic collision, where the bat is much heavier than the ball, the ball will be driven at twice the speed of the bat.
If the bat were the same weight as the ball, then with an elastic collision, the bat should stop completely, and the ball should be driven away with the speed that the bat used to have.
If the bat were lighter than the ball, then the bat should rebound, and the ball would be driven with less speed than the bat had originally.
Since we know that neither baseball bats nor golf clubs rebound, we know that the ball will be driven faster than the club head was travelling, but not twice as fast (as the clubs aren't infinitely heavy).
A golf club has more of its mass concentrated in the head, and it's longer and lighter, so I would expect the golf club head to be going about twice as fast as a basball bat head, if the same amount of energy is put into it.
So I expect a clear victory for the golf club.