I know this is REALLY late, but if Batman could beat Superman why would Superman even bother giving Batman the kryptonite ring. Also, Batman may cheat, but he wouldn't kill Superman, if the fight was no holds bar, everyone cheat/kill/bite as much as they want, Batman would be dead...er faster than a speeding bullet?
Maybe he trusts Batman to use the ring in case Superman becomes evil or otherwise a threat to humanity. Not saying that Superman would ever do that, but in case a supervillan hypnotizes him or takes over his body or something.
And as long as we're talking about the animated Batman, remember, he does carry a piece of kryptonite in his utility belt for "insurance".