Essentially, you're doing the same thing which is separating the biological aspects from the socially constructed aspects. Rather than Sex vs Gender you'll have

(Sex/Gender) vs (Sex/Gender roles, sterotypes, behaviors)

Either way, you're making the same distinction, so it's really a matter of semantics. I think Sex vs Gender seems to be less confusing, simply because TRAs have already rallied around the term "gender" with terms like transgender, gender fluid, gender identity, gender dysphoria, whereas LGB utilize "sex" in all our terminology (homosexual, bisexual, heterosexual, same-sex marriage, sexual orientation). The debate has already been framed as one of Sex vs Gender. Now it's time to be gender critical and make the argument that sex is more important than gender.