On Final Crisis: Secret Files, Grant Morrison describes Doc Fate as part of the Society of Super-Heroes, and says that amoung it's members is, sometimes, "the mysterious Bat-Man". Obviously, this Bat-Man haven't appeard in Multiversity. But how do you think this pulp version of Batman should be? It thought of him as Kirk Langstrom, the Man-Bat.
It being Kirk Langstrom could work. Maybe Bruce Wayne could exist in this world but he's not really Batman, maybe Batman's agent or secret identity or the source of Langstrom's money. I'm not particularly interested in Man-Bat as a character but the idea of him being this world's Bat-Man, or at least "a" Bat-Man, could be a good idea.
I find most versions of "Batman but he's pulp" to be generally really boring because they largely just go by the painfully standard route of "Batman but his outfit looks shabby and he's got a gun" and it often seems like it just amounts to a Batman that now kills and is less of a character than just an amalgam of pulp cliches (which is what Batman used to be before they created Robin and the iconic villains and the no-kill rule and started fleshing him out as a more unique character).
In fact I'm gonna just go ahead and stipulate that "Pulp" Batman should at least adhere to the main dude's policy of no guns. Whether he's killing or not, or how effectively he can manage himself, is up to debate, but let's get the obvious out of the way. Bill Finger described once that he envisioned Batman as combining traits from Douglas Fairbanks, Sherlock Holmes, The Shadow and Doc Savage, so we're gonna use these as some of the basis to work with.
Personally the best take on a Batman redesign schewing closer to pulp that I've ever seen would be "The Batman Investigations" by Imson on Deviantart, you can read more about it's respective designs there but I'll post it's idea for Batman below
Bruce Wayne was born with high sensitivity to sunlight, keeping him indoors most of the day, and forcing him to wear tinted glasses when not. When his parents were killed by a mugger, the blast from the gunfire damaged his hearing, forcing him to wear hearing aid from an early age.
Bruce had a knack for mechanics, and living off the remaining funds left by his parents, he developed a few surveillance gadgets for police use. While trying to market them for the local police, he was hired by commissioner Gordon as a freelance detective. As his involvement with the force continued, he got the nickname The Batman from his striking appearance.
As a hired gun, his role grew to take on undercover investigations. He could use his equipment to take the cases further than regular detectives could risk, and the police would turn a blind eye to these acts, provided Bruce denied any involvement with the police should he get caught.
His most useful gadget is his mechanic surveillance bird dubbed Robin, used to keep watch on suspects without gathering attention.
This one I think works particularly well for a take on Batman that's more centered around street-level detective work, and I mean actual detective work. A Batman who's strong, but not much of a fighter, and his hearing disability doesn't lend itself to superpowers, it's something he has to work around through allies and gadgets and quick-thinking once he starts getting further into trouble. More of a Sherlockian Batman who contends with the kind of darkness that doesn't come with a status quo to sort itself out.
Now, if we wanna go more along the route of gritty pulp Batman, a more traditional Batman who's actually gonna out there in the night suplexing crime into the sidewalk, what comes to my mind design-wise is something of a cross between Denis Medri's Rockability Batman and Francesco Francavilla's Batman 1972
A Batman who's not exactly wearing a skintight superhero outfit full of gadgets but instead a coat of surprises, who's only really got a mask and goggles to protect his identity and also protect his face. A Batman who's smart and can do detective work, but really he's going out there every night to get the shit kicked out of him until he figures out what he needs to know to solve the case, no police involved. Sort of closer to how Denny Colt or The Moon Man operate, more in the Robin Hood and Zorro molds. This one is also somewhat in line with Golden Age pre-Robin Batman where his main distinguishing feature as a character was how utterly reckless and prone to jumping into a fight he was even by usual comic book/pulp hero standards.
The third option I'm thinking of is to take Batman closer to the superhero he is but still mired in enough pulp ingredients remixed differently to be recognizably different from his ordinary self. So instead I'm thinking to play up respectively the parts of Batman that are The Shadow + Doc Savage, to create Dark Science Batman. This one I think is a better answer as to the question "What would a fusion between The Shadow and Doc Savage look like" than my initial response of a depowered Black Terror, and this Batman would be considerably more inhuman and grandiose in scope than the other two, this would be a Batman who could dominate Justice League meetings by force of personality and create moon bases and contend with dimension-warping supervillains and whatnot.
This one is inspired by a redesign by Arryn Diaz and Dan Schkade's Jack Kirby Batman:
“In a WORLD gone MAD, this lone DETECTO-SCIENTIST must operate OUTSIDE the laws of society in order to defend its FUTURE!! BEWARE — THE BAT-MAN!!!” - Alternate Universe Jack Kirby, 1972
So these are three different directions as to how I think you could do a "Pulp Batman" well, although maybe not necessarily operating by how Earth 20 works. DC already has a billion zillion Batmen as is so I'm sure all these may have already been done at some point in Batman's history.
Of course, realistically speaking if Batman was a long-running and successful pulp hero he could have gone through variations of these depending on the decade, or the country (see: the many global variations of characters attached to the name Fantomas or Zigomar), if not the same narrative arc outright. Pulp heroes are shapeshifters by nature.
