Isn't interesting that some people who grew up on these cartoons would today say that those force "LGBT ideology" on kids, if those exact same cartoons were produced today?
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.
They will say that this is not the same (kind of like Vance has old pictures being dressed as a woman), but if Bugs Bunny was released today it would be labeled woke.
Drag queens are doing this for comedy, that's why their makeup is over the top and that's why many have a massive beard.
I believe you are overestimating the abilities of mainstream online transphobes to critically examine a piece of media beyond "boy wear girl clothes???"
SHOOT
They don't have to shoot you now.