"John Dillinger" (b/w "Kidnaping is a Terrible Crime") on Bluebird 5522 (1934), by Joe Smith, the Colorado Cowboy, BKA Dwight Butcher. I think this is on the 70s reissue of his old recordings but don't have the LP at hand to check it. Butcher also did the first "Bonnie and Clyde" song, likewise in his Joe Smith persona.

I'm also thinking of a "Goodbye, Public Enemy #1" song from those days, though I can't find anything under that title in Tony Russell's country music discography and this may just be the chorus of Butcher's "John Dillinger," which I haven't heard since Hector was a pup.

I've never heard it, but there is also Champion 16790/45046, a 1934 recording by the Lone Star Cowboys (not Leon Chappelear's group of the same name) called "Dillinger's Warning."