My immediate response is "Eh."
Here, it's portrayed as a joke in a way that is presented such that the humor is derived from how obviously Bugs is not a woman and is the same rabbit that he's hunting, and that Elmer is really really dumb for falling for it. I don't believe that transphobes would decry this depiction, even today, because it aligns with their sensibilities.
Oh, that was all my own take from an external observer's viewpoint.
They would likely just stop at "bugs bunny = good because I like it" and refuse to engage critically past that.