Uhm
Because apparently...
... holding the belief (in principle, at the very least) that transmen should be allowed to use the men's changing rooms (and vice-versa)...
... is "misogynistic" and/or "anti-woman", and/or "sexist".
And since the person behind this claim states that "Being pro-trans doesn't necessarily mean being anti-woman" and since the person in question has also stated that they are (by their own estimation) pro-trans,
this therefore means by direct inference that the "correct/acceptable" concept of being pro-trans (in this person's eyes) cannot include the holding of a belief that transmen should be allowed to use the men's changing rooms (and vice-versa).
So apparently it's all good to be pro-trans - and aren't we all pro-trans (or should that perhaps more accurately be: we must avoid giving any indication that we're not pro-trans...) - but only if it's a certain type of pro-trans belief. One which denies something which is pretty central to the very core of what transgender identity means at both a philosophical and practical level.
"Oh yes, of course I'm pro-trans! (Except I want to force transmen to use the women's changing rooms, and vice versa.)"
There was some ninja-ing going on there. Now i see more clearly the error in your logic.
I'm going to make some assumptions about some positions here, as I express them in different words. Please correct me if I am in error about your positions.
Your position is that it is impossible to be pro-trans unless you support changing rooms that are usable by all men, including trans men, and vice versa. (I'm trying to stick to my newly adopted vocabulary here. "Men" is people with a masculine gender identity.)
Emily's cat's position is that it is mysogynistic to allow a male, including a transwoman, to use a changing room that was intended for female use. (Such a changing room under the archaic language would be called the "women's changing room")
More formally:
Your proposition is "If you do not support trans access to their choice of changing rooms, then you are not pro-trans,"
Her proposition is "If you allow males into a female changing room, you are misogynistic."
You then mix it up. You take your proposition about what it means to be pro-trans, and you combine it with her proposition about what it means to be misogynistic. So you are mixing propositions from two different arguments.
You have imposed a third, previously unstated, proposition on the argument, that if you are pro-trans, you are not misogynistic.
From a persopective of logic, your "logical inference" doesn't fly.
In addition to being an invalid logical inference, several of us disagree with your proposition, quite independent of it's use in the arguments. I believe it is possible to be pro-trans and still reject unrestricted access to private female spaces.
I also will say that I agree with EC's proposition, or at least with something very close to it. I say that if a majority of women want something, and you say it is not important, then you are misogynistic. In saying this, I am adding something to EC's statement. I am adding that keeping males out of female only spaces is something that the majority of women would want.
(Yes, I was careful about word choice there, regarding "female" versus "woman".)