Eh, what I'd probably do would be more an Elseworlds thing, I suppose, but a few things I'd do:
* Bruce's origin would be pretty much the same, except during his training abroad he also picked up acrobatic skills by training with the Flying Graysons, befriending John - this later sets up Bruce helping bring their act back to America for their fateful final show (and also provides a measure of guilt for Batman, resolving to bring their killers to justice in spite of Zucco's hold on the investigating officers). As a momento of his time with the Graysons, he keeps in the cave one of their first-generation costumes (the traditional Robin suit).
* Gordon learned Batman's true identity years ago, but has been in a coma since as the result of a bust that "went bad" (translation: a double-cross). The anti-vigilantist Ellen Yindel is the current Police Commissioner, which is good for Gotham's underworld, since it enables them to entrust Gotham's 'finest' with a mutual primary objective: get Batman, dead or alive. With Gordon out of action, Batman's only remaining confidant on the force is Harvey Bullock, who has been working both sides of the line to his advantage until Batman blackmailed him into his service. Bullock naturally hates his guts for this.
* Robin is Barbara Gordon, wearing a version of the first-generation Flying Graysons outfit; and Dick Grayson gets skipped straight to Nightwing after resolving to avenge his parents' murder by wearing a darker version of the second-generation costume. Batman reluctantly takes Robin as his protege, and keeps an eye on Nightwing, who insists on working solo; however, both know Batman's true identity, and in addition to Alfred the only other person who knows is Dick's aunt, Harriet Cooper, with whom Dick lives but who also has access to the Batcave, and is a saucy old broad who can give as good as she gets; at one point she even manages to fend off Killer Croc after he finds his way to the Batcave (though not knowing Wayne Manor is directly above it) while disguising herself in one of Bruce's old capes and cowl, earning the honorary distinction of "Batgirl".
* As far as Batman's general attitude and relationship with his allies, he's probably somewhere between Batman Beyond and TB&TB Batman - what sense of humor does crack through from time to time is a bit snarky and maybe a little sadistic. Has a tendency to call Robin "Old Chum" just to aggravate the heck out of her; he claims he's trying to teach her the value of self-control - she believes he's just being a creep.
* Victor Friese is forever sealed in sub-zero temperatures inside a reconditioned old NASA spacesuit with a supercharged ice-ray housed inside a snowblower that can freeze anything or anyone solid in seconds. His face is never seen, and he can only speak through a voice box a la Dr. Phibes. He keeps
what's left of his wife in cryogenic storage hoping to reverse whatever it was that happened to her (here's a hint: think "Futurama" or even TNBA for a second), and in the one moment where Batman and Robin DO get his helmet off of him in the thick of battle, what they see makes Batman gape in horror and makes Robin sick to her stomach...suggesting that Freeze may really be beyond any sort of physical help.
* The mixture of chemicals that the first Red Hood fell into that turned him into the Joker included those used in embalming fluid; the mixture has caused his body to thin out, his hair to yellow to an almost greenish hue, and his lips and eyelids to peel back rendering him a horrible, gaping, grinning corpse-like figure...but it has also granted him enhanced speed, endurance and reflexes. Has a tendency to use a high-powered pogo stick which he maneuvers with all the mastery of a champion equestrian.
* Killer Croc's skin condition basically makes his body look like a giant callous, and its smell isn't helped any by his insistence on stalking the sewers to "dehumanize" himself and make himself every bit the intimidating shadow figure that Batman is. He's even gone so far as to have his toenails, fingernails and TEETH sharpened to deliberately make himself a more monstrous figure, and has also honed his ability to hold his breath underwater for significantly long periods of time.
* Batman has "The Car", Robin has a motorcycle, and THAT'S IT. The car does have amphibious capabilities and short-range rocket boosters (a la "Knight Rider"), otherwise it's not an elaborate-looking automobile - just a jet-black street vehicle that rarely makes use of its headlights. Aerial recon is usually done hang-gliding from airfoils, otherwise from rooftops.
* Yindel is convinced that Bruce Wayne is Batman, but has yet to be able to prove it. Most of Gotham's underworld, however, has a hard time swallowing the snobbish fop Wayne living a dual life as a guy who breaks bones and knocks out teeth regularly every night.
* The Batcave is powered by an ore Bruce had synthesized from a peculiar meteorite that was first recorded in North America 30 years prior, one that emitted low-level radiation that could prove poisonous to humans with prolonged exposure
; naturally he, being Batman, has taken precautions.
