I know this isn't an especially academic answer, so it might not be ideal for this sub, but because this is a question about fictional characters and the intents of their authors, and I have some familiarity with the subject, I thought I might at least put an answer out there.

For a little background: Dick Tracy comics debut in 1931 and it typifies a type of character and drama that was already present in media and becomes quite popular. The detective fiction trope has a lot longer history, but the Dick Tracy strips became especially influential in the genre of periodically published illustrated stories. Chester Gould who wrote the strip was interested in police procedure and made an effort to study some contemporary methods of police work, but he wrote improvisational stories loosely based on his idea of how actual police might engage in things like forensic work. He variously claimed that Tracy stories had been based on real-life New York police officers and US federal agents, but those connections were loose. His sensationalized villains like BO plenty and Giant Setton were based on tropes that had already appeared in fiction. Some of those tropes trace themselves back to the detective fiction of Dashiell Hammett, who claimed to base much of his writing on his actual work as a Pinkerton employee. Some tropes like the 'midget bandit' trope of Giant Setton can be found in Hammett's work, and he claimed that these were based on actual criminals he had helped to apprehend, some writers like Michael Fitzgerald of the Stockton Record have tried to investigate some of Hammett's claims with limited success.

One thing we learn if we look into research on Hammett and his claims, is that newspapers historically loved a catchy headline. Fitzgerald's investigation uncovers headlines from regional newspaper archives including 'the gentleman bandit', 'the hermit bandit', 'the boy bandit', 'the miner bandit', and others in his search for the midget bandit.

Detective Comics launches in 1937, Batman doesn't appear until issue #27 in 1939. Detective Comics is once again loosely based on real police work but mostly characterized by simple tropes common to the genre. Early villains in the Detective comics series include Ching Lung who we now characterize as part of the 'Yellow Peril' stereotypes, which structurally were making Southeast Asian ethnic groups into mostly generic villains. Ching Lung, like BO Plenty (and the associated 'Yokel' villain trope), were not so much based on 'real criminals' as they were based on real social and ethnic prejudices that were pervasive throughout media, not just hard-boiled detective fiction. Criminal syndicates organized by Asian immigrants existed, and illiterate poor people did lie to police officers and get arrested for it, but the characters weren't drawn from a specific arrest so much as they were drawn from a collective vision of a type of person who commits crimes and becomes a villain to the author.

So here we can address at least part of your question. Did actual criminals of the era have over-the-top gimmicks, motifs, or theatrics? Not so much, an actual look at claims made by Hammett and others which appears to be based on real police work reveal a lot of media sensationalism. Regional newspaper investigations have identified historical local criminals from the 20's and 30's described as 'the midget bandit' whose actual height was between 5'3" and 5'5". So it is true that the Pinkerton agency did assist in identifying a killer who a local community had identified as 'The Midget Bandit', but it was really just a criminal who wasn't especially tall, and this makes its way into hard-boiled detective fiction as Giant Setton who is depicted between 3'- 4' tall or smaller.

Batman villains borrow a lot from earlier horror fiction. Clay Face is introduced in 1940; artist/author Bob Kane claims that Clay Face, a disgruntled actor who gets fired and seeks revenge while dressed as a character from a horror movie he acted in, was based on Phantom of the Opera. Phantom of the Opera is itself entirely a work of fiction, but scenes from the film like the falling chandelier sequence are related to real events like the chandelier counterweight falling through the Palais Garnier (which appears to have just been an accident). Catwoman is based on actress Hedy Lamarr rather than a criminal, Kane saw cats as a thing that would hunt and play with bats, and wanted his comic to have sex-appeal, she is well aligned with the 'Gentleman Thief' archetype. Hugo Strange is introduced in 1940 as a generic Mad Scientist trope, generally traced back to Frankenstein, and in later plots he creates devices that strongly mirror themes in Dr. Jekyll. Mr. Freeze is originally introduced in 1959 as a joke villain, his eventual minor similarity to the murders of "Iceman" Kuklinsky in the 60's and 70's is coincidental. The post-39 Detective Comics issues are pretty heavily influenced by films rather than real life, Kane references films frequently in his writing/interviews about his inspirations. It may also be notable that early batman villains like Joe Chill didn't chill their victims, they were just murderers with catchy, entertaining, or identifiably villainous names and no real trope or gimmick ('Chilled' appears in detective fiction as a slang for 'murdered' in the 1920s and 30s).

Detective fiction is at times exaggerating characteristics which had already been sensationalized by local reporting about real crimes. This gets mixed with the prejudices and inspirations of the authors to create fictionalized villains which now lack strong connections to real life criminals. I've seen a lot of loose connections made to real criminals as the genre continues. Sometimes people point out that The Penguin is eventually portrayed as using a Bulgarian Umbrella, a pneumatic weapon hidden in an ordinary object, which was used in an actual murder in the 60s - but The Penguin was originally introduced in 1941 as another Gentlman Thief archetype who uses a cane gun to murder a mob boss, cane guns had been associated with the Gentleman Thief archetype since at least the 1920s, and real cane guns had been produced and used for decades before that. So once again claims to a connection to a real 1960's murder are coincidental, and later penguin plots reflect or embellish this coincidental similarity.

This doesn't necessarily address the broader question 'were there ever any especially gimmicky criminals.' However, it might address the related question 'were these specific Batman villains originally designed based on real-life equivalents' and the answer to that is mostly no, the tropes are works of sensational fiction with only loose connections to real practices or criminals. Iceman Kuklinski and Georgi Markov were kind of sensational real-life villains later associated with an elaborate plot or gimmick, but the connection between these criminals and later comic plots, or between real-life 'midget bandits' and the tropes of detective fiction in the 30s and 40s is like many connections between fictional portrayals and real life, inspirational but not directly influential.