Maybe it's more like their belief in women being inferior fuels their transphobia and vice versa

this is actually a really good way of thinking about it, it's called transmisogyny theory. basically transphobia is motivated by both the idea that masculinity is better than femininity and the idea that male and female are exclusive categories. when someone breaks that boundary it's always met with hostility, and doubly so for people who are moving into the "inferior" category. i just started julia serano's whipping girl and it's very good and gets into far more detail on this than i or the wiki article can