The apparition of impostor!Lavender Jack make me wonder : do you know if the "I'm like you but evil" villain archetype was popular in pulp fiction or is it more a super-hero thing ?
It's very much more of a superhero thing. Not that it didn't exist before, obviously the idea of villains designed to resemble and contrast their heroes is as old as villainy itself, but the idea of a supervillain who's specifically meant to be an evil version of the superhero, the "Inverted-Superhero Supervillain" as Peter Coogan calls it, was defined in comics.
If you wanna get specific, technically the first supervillain to be specifically defined as an evil opposite to the hero (as opposed to just being an evil take on a general heroic concept) was Moriarty, who's is very strongly defined as almost an evil twin of Holmes. This, I'd argue, is Moriarty's greatest contribution to the history of the supervillain, because he was neither the first, nor the one who popularized the idea of a supervillain or arch-enemy (those would be Dr Jack Quartz of the Nick Carter magazines as well as the grand criminals from the feuilletons that inspired Holmes).
What the pulps had, in turn, was supervillains who were meant to evoke popular heroes, like Fantomas who evokes the gentleman thief and John Sunlight in his original form who greatly resembles Holmes, and supervillains who were protagonists, but not specifically inverted takes on superheroes, because those as we define them weren't around. The Shadow fought several criminals who were intended to evoke him, and as far as I can find Gibson was the first person to specifically coin the term "super-crook/criminal/villain" to describe villains (which does not mean he created the concept).
The grand criminals of the dime novels and feuilletons led to the pulp supervillains, which grew bigger and badder and more outlandish and laid down much of the foundations of what we currently used to define supervillains. And throughout this history, the idea of costume-wearing supervillains gradually starts to show up, first of these being the Wolf Devil from Queen of the Northwoods (1929), likely the first superpowered costumed supervillain in Anglo media, followed by the Klan robe-wearing pulp villains, and then odd costumed supervillains like Bill Everett's Great Question and The Lightning from The Fighting Devil Dogs, until at last we get to the comic book supervillains proper. And with them, the Inverted-Superhero Supervillain, as Peter Coogan describes it:
The inverted-superhero supervillain is limited to the superhero genre, primarily because they have superpowers, codenames, and costumes. Although there are earlier costumed supervillains in comics—such as the vampiric Monk, whose schemes Batman ruins in Detective Comics #31—the Joker and Catwoman are probably the best, early examples of inverted-superhero supervillains.
Prior villains like the Monk draw on masked and robed pulp predecessors and mad scientists like Lex Luthor or the Hugo Strange have a long lineage outside of comics. But the Joker and Catwoman mark an innovation in villainy because they are such direct responses to the superhero by creators looking to expand the superhero genre.
It's a bit trickier to say which exactly would be the first Inverted-Superhero Supervillain, along the lines of what you describe. Coogan claims it's the Joker and I disagree, because while the Joker's contrast with Batman was definitely important to his popularity and he represented a clear break away from the more pulp-esque Monk and Hugo Strange, he was hardly intended as an evil version of Batman (that would be Killer Moth in the 50s), nor was he that different from the common Dick Tracy villains or other villainous clowns in fiction like The Whisperer's Grim Joker or The Shadow's Number One to really merit that kind of distinction. You can point to other Golden Age supervillains who specifically take the superhero image of "caped man with a chest logo and/or cape & mask" like Fawcett's Captain Nazi or MLJ's Captain Swastika. I'm fairly sure there's earlier examples still, probably in the funny animal superhero comics that influenced Fawcett's output, but I'd have to go digging further for those.
Although I will point out that The Grey Claw, who I discovered more recently actually debuted almost a year ahead of Superman (in 1937), was a comic supervillain very much dressed in what nowadays we'd consider a superhero outfit, and also predating the Lightning's own costume. He was modeled quite a bit on The Shadow, not just in costume but also in laugh and mannerisms and radio boogeyman persona, but he was armed with weird sci-fi weaponry at odds with the gritty crime story he's in, and he was dressed in a costume that included not just a cape and slouch hat, but also a mask, a waistband sash, and a chest logo, and even did the classic Superman pose as seen above. Is he the true first Inverted-Superhero Supervillain? Probably not, despite his international publishing, history has not been kind to him. But so far, I'd say he's as good as any candidate.
I'm totally not biased on this regard, though. No, sir.
