I'd imagine if Batman has feats with different pieces of gear that he would implicitly have that gear if you're allowed to use every feat. The OP said Composite Batman and composite usually means all versions combined -- this would include the brief, or even non canon stuff (Like movies, as if that mattered). I don't get why people are being sticklers about the OP not saying "Batman with lantern rings" when Batman with Lantern rings would be one of many things included in Composite Batman's arsenal.
If the thread creator truly meant a Composite of all versions of Batman and of all feats that's ever entailed, I would think that Batman probably wins. Nothing Saitama's ever done really even scratches Lantern abilities besides maybe combat speed.
This is ignoring Karate Kid who would probably give him an amusingly good fight on his own. Karate Kid makes a joke out of Kryptonians and Daxamites, who are quite a bit stronger and similarly or more durable than Saitama. While those two categories don't seem to fight as fast as Saitama(atleast on average, I think Superman is presumably capable of fighting at and above Saitama's speed), they're awfully close and Karate Kid has pretty silly feats (he's deflected meteors, too! With much less collateral damage!)
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