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CIS and PIS on, in character.

Batman gets a gun. No h2h allowed, this is a shootout.

depends a lot on the environment i think. standard ring, open ground, punisher probably gets him because he'd go for a kill more willingly. bats is faster though so it would make frank's job harder and he'd likely take a bullet somewhere non-lethally first. in a city, bats can sneak up on him too easily and incapacitate him. if frank can still shoot though bats could still lose. bats would never kill him, but he'd need to ko him otherwise frank will just keep going for a kill until he gets it.

Would Frank kill him?

Thought he had a code against killing heroes, which is why he used rubber bullets against Matt and Spidey.

This seems a standard forum fight (unless stips start being added to appease the Bat-gods), half mile start distance (**** the metric system), in a featureless environment with handguns.
Effective range for a 9mm (where a shooter can deliver a hit to a man size target) is roughly 5,900 feet, a half mile is about half that give or take so they can engage immediately.
One is more used to dodging bullets but less experienced at gunfights, both have armor...i'd go with the better shot.
That's Frank.