If the prevailing notion is Bob is playing dress up as a woman I don’t think there would ever be a problem with the right. The left would never agree with that.
That's not what I got from the parent. Besides, everyone is dressing up one way or another depending on the situation. The problem is putting all your identity into it. And conversely also imposing on others that some characteristics you find important in your own belief system should be part of their identity. Extreme left and extreme right both have issues with that.
I think it depends on which factions of the right and the left. As I understand it, left-wing radical feminists mostly already hold that view. And some on the socially conservative right may still object to Bob/Brenda teaching their children, for example.
However I do believe this position, or one very similar to it, could be enough of a middle-ground compromise to satisfy most people.