I know you guys don't like crossovers, but Batman's detective skills can be pretty extreme like against Terminus, and he's dealt with Predators successfully before. As far as Captain America, he's not only far stronger, durable, and faster than Batman but he's a better fighter and never gets tired.
Also, if all it took was a gun to kill Batman he'd have died a long time ago. He may not be faster than the bullet but he only has to be faster than the shooter can aim, and unless it hits him in the mouth it'll hit bodyarmor.
As far as h2h, c'mon, it's Batman we're talking about.
That said, I do not know enough about Mystique to say who would win.
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