In 1894, Thomas Edison's movie studio Black Maria filmed a brief, wonderful short, featuring two cats boxing:
The Library of Congress explains that this match was part of a whole cat-boxing thing:
"The performance was part of Professor Henry Welton’s 'cat circus', which toured the United States both before and after appearing in Edison’s film. Performances included cats riding small bicycles and doing somersaults, with the boxing match being the highlight of the show."
It's hard to find more information about the Welton cats outside the context of the Edison film, but it's easier to learn about another pair of fighting cats who featured in the 1937 film Something to Sing About:
According to a 1937 issue of the Eugene Register-Guard, the cats Pinkey and Pal were trained by "expert feline psychologist" Arthur Nelson. The paper described their performance: "Pinkie (sic?) and Pal went through their act in a fighting, slugging flurry of furious action, and both of them evidently enjoyed the whole set-to," and noted that the Los Angeles ASPCA approved the shoot.
It's unclear what the ASPCA would have said about the live-action cat boxing show at a "New York night club," refereed by Arthur Nelson. From Popular Science's September 1937 issue:
Danielle Wiener-Bronner is a news reporter.