There were unquestionably subtle homo references and allusions back then, especially in the pre-code (pre-1934) films, before the Hays code. However, I don’t think that was the case with the example you cited, as “gay” would have been meant in the wacky/balmy sense... a guy in a woman’s bathrobe being akin more to waking up after a night’s wild-party, drunken silliness like someone wearing a lampshade for a hat.
A woman’s bathrobe as a humor device (like the time in the “Blondie” comic-strip in which the doorbell rings and Dagwood jumps from the bathtub and into Blondie’s bathrobe to answer the door... prompting the salesman to say “what a homely dame”) pops up every now and then, and generally wasn’t a ‘gay’ gag.
In the context of the scene, it would seem that when the Cary Grant character said, "Because I just went gay all of a sudden!" it would seem not to mean “gay” as in “happy or carefree” but in the other sense as it was in answer to why he was wearing a frilly woman’s robe, but it could be taken that way as well. But then it would have made more sense to say “I was just feeling gay all of a sudden”.
Bringing Up Baby - "Gay All of a Sudden"
The word gay was long meant to mean "joyful", "carefree" or "bright and showy" but it did have a more negative usage going back to the late 1600’s to mean uninhibited or without moral constraints, loose or promiscuous and could apply to either men or women, presumably straight. The use of the word gay to mean homosexual, goes back to the late 19th century or at least the 1920’s but wasn’t in much regular usage in that way and was more often still associated with the earlier meanings of being happy and carefree until the 1960’s.
I grew up in the 1960-70’s and even as a kid, I knew that in a certain contexts, “gay” meant homosexual or as my father used to refer to them; “light in the loafers”, “poofers” or “Nancy Boys”, even if back then I didn’t completely understand all of what that meant, I did understand it to mean a man who was effeminate and preferred the company of other men. I also knew that when the Flintstone’s theme song said “We’ll have a gay old time” they didn’t mean that Fred was hooking up with Barney and Wilma with Betty.