There is certainly a longstanding cultural thread of defending traditional manhood through explicit castigation of male deviants, but notably the source of the explicit castigation is largely from other males. The pushback against trans-women's acceptance as women isn't largely driven by men. It is pretty evenly distributed, or perhaps even more driven by women. The point is that these seem to be distinct phenomena driven by distinct concerns.

I don't have any hard numbers, but it is the impression I get from various organizations and legal challenges to trans legislation in the UK. The organized opposition seems to be largely driven by women.