I will be the first to admit I know little in the way of food, 'specially fancy stuff like what you mentioned... but, I will throw myself in here and give it a shot. And while The Bat Cave mighty be a little... straightforward, perhaps
explicit, subtleties are often lost anyway on young children, and it might be way too fun to be able for parents and kids alike to say "TO THE BAT CAVE" when pondering where they'd like to eat. Or maybe that's just me. Anyway.
Hmm... Gotta involve the Bat-Mite somewhere, like it or not. Prolly on the kid's menu. Maybe the unofficial mascot? But that's not a dish, need to think of a dish.
Commissioner Gordon ought to have a hearty
but comparatively bland Cracker Barrel, southern-style breakfast old folks like to eat. Pancakes, sausage, eggs. That sort of thing.
Dick Grayson. Perhaps focus less on history as Robin, more as the Flying Grayson's? Menu entry should contain "Holy _____, Batman!" Food... nothing coming to mind.
Killer croc, definitely seafood.
Or chicken.Mad Hatter? Can't think of anything but tea. Come back to later.
Poison Ivy? Vegetarian dish.
Bane... probably a steak? Big steak.
Joe Chill's icecream? Too obvious. Some kind of frozen food... No, Mr. Freeze is better.
Ah!
Utility Belt shish-kabob?...

This is way harder than I thought it would be! I know all this crud about Batman and friends, but I have
nothing on food. I'm sorry. I'll let you know if I come up with anything else, and maybe I'd be better at helping you write menu entries, but... My knowledge on restaurant dining is horribly limited.
...Harvey Bullock gets a burger, though.