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Who wins

Doom probably has better power output.

Perhaps. Bat-Mite's best feats are casually killing Spectre and erasing a universe.

Thing is, Bat-Mite tanked an attack from Mxy that erased the whole of DC: all pasts, all presents, and all futures... Literally everything... The entire damn company.

IOW, even if God Doom's output > BM's, Doom still doesn't have a way of putting Mitey down.