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

i mean that's an added side benefit for these people, but phrasing it like this kind of makes it sound like they're enforcing transphobia just to hurt cis women. it's very much the opposite! cis women uphold transphobia almost if not exactly as much as cis men, and when we spend too much time emphasizing the fact that transphobia harms them as well (despite being obviously true) it erases the fact that the true target of this is always trans women