I'm not certain the hairdresser was "gay." I'm thinking that Bugs was playing the stereotypical "female hairdresser" (as if "in drag") or something. Bugs was most excellent when "in drag," when he felt the situation warranted it.
I love to point out the mouse BS, but I’m not sure this is it, this isn’t making one’s kids anymore gay than clinger in MASH.
Infact I think MASH wouldn’t have been able to have been made today, showing crossdressing in a funny way and as a way for someone to try to be sent back home