Two things:
1. TDKR has a tank, and that's one of the most influential Batman comics of all time. So it was probably an inspiration and it probably has roots (it was originally built as a riot suppressor, so it has more similarities to the game's version than you seem to think).
2. The entire point of Batman is that he's prepared for every single outcome, every eventuality. Ever read the story where he has to take down the Justice League? One of the first things he did when he joined was prepare a way to take them out if something required it.
The city is attacked by tanks. Batman will have something to use against them, that's the entire point of his character. He's a boyscout with a vengeance.
So yes, it fits the game and universe snugly.
It really doesn't. None of these things happened spontaneously. "The city being attacked by AI tanks" almost certainly came after the battle-mode. So, like I say, it's a contrivance.
My point is that the Batmobile operates separately from being a tank. It pretty much always has. In one instance, in DKR, it was a vehicle designed to suppress riots. In another instance, Batman Begins, it was an armoured vehicle with munitions intended to destroy barricades. In Batman: Arkham Knight it is a war machine.
I'm not saying you can't do it. It only feels slightly out of character for Batman to be driving a minigun equipped tank. My point is that you'd be more faithful to what the Batmobile is if you had a vehicle designed without it. So the tank-mobile happened for gameplay reasons, and ones that I think became tiresome after the first few instances and certainly shouldn't have taken the limelight from predator or even pursuit-mode sequences.