> Perhaps it would then also be helpful to use toxic misandry instead of toxic masculinity

Toxic masculinity isn't misandry, so, no, it wouldn't be helpful. Both toxic masculinity and internalized misogyny are aspects of partriarchal socialization; they aren't symmetrical with regard to sex or gender.

Toxic masculinity seems straightforwardly like misandry to me. Someone saying "you're not a real woman unless you're skinny" would obviously be considered misogynistic, so someone saying "you're not a real man unless you're muscular" seems straightforwardly misandristic.

I agree, toxic misandry or internalized misandry would be much clearer.