Does Your Identity Reinforce The Gender Binary?
First, I'd like to acknowledge that I'm weighing in as an outsider. As a cisgender aromantic asexual, none of the labels I'm discussing here relate to me personally. The closest I come to identifying as any of these labels is ticking 'bisexual' in surveys that provide only three sexuality options, because my own attraction is equally nonexistent for all genders.
But it's interesting to me that I don't see the same furor over lesbian-identified women dating trans men. (Or the reverse, though I don't see that very often at all.) While some women choose to change their identification because of the realisation that they're also attracted to trans men, others don't, and it seems to be recognized as a personal choice and not necessarily a political one.
And I agree with that stance, by the way. After all, there are women who identify as lesbians despite experiencing some attraction to cis men, too. A woman who is Kinsey 5 may feel that lesbian is the best descriptor for her identity, despite her potential to be attracted to men. Or a woman who is biromantic homosexual or bisexual homoromantic may identify as lesbian because she only feels both attraction types together in relationships with other women, or because her attractions counted together result in a significant preference for women. If a woman can have attraction for cis men and still be lesbian, certainly a woman can have attraction to trans men and still be lesbian.
Personally, I don't think you can judge from someone's label, or their label plus their romantic and sexual choices, whether they support binarism or transphobia. It's the reasoning behind their label choice, and their overall behavior and attitudes towards transgender and nonbinary people, which really determine whether they're binarist and/or transphobic.
For example, there was a straight man I heard of who dated a trans woman, but insisted that she had to allow him to have casual sex with cis women in order to meet his sexual 'needs' because he was 'not gay' and therefore not satisfied by sex with his non-op partner. His labelling himself straight while dating a trans woman superficially sounds accepting, but the fact that he insisted that she couldn't satisfy him sexually because he wasn't attracted to men belies his apparent acceptance of her identity, and suggests that he really did see her as a man on some level.
In contrast, my brother has a male friend who started dating someone he believed was female. His partner then came out first as FtM, and then they came out again as nonbinary. In response, he shifted his own identity from straight to bisexual. Although I haven't asked him about his feelings in depth, from my understanding, he has never been attracted to cis men. His identity as bisexual appears to be based on his potential to be attracted to cis women and transgender/nonbinary individuals, in which case he clearly demonstrates a willingness to acknowledge trans and nonbinary identities using a label commonly stereotyped as binarist.
I have also heard of bisexual-identified people who say "I have never knowingly met a nonbinary person, so how can I tell if I'm attracted to them?" Exposure to cis males and cis females is pretty much universal, so a questioning person can readily collect data on how their sexuality reacts to these people. Transgender and/or nonbinary people are a lot harder to find, and may not be out when you encounter them.
There are also nonbinary and transgender people who identify as bisexual. And there are nonbinary and/or transgender AFAB who identify as lesbians, and AMAB who identify as gay, despite those labels implying a gender they don't identify with. (This seems especially likely for nonbinary people who exist in the spectrum between cis and gender neutral, such as AFAB demigirls, but there are FtMs who consider themselves lesbians despite identifying completely as male.)
In conclusion, don't judge from the label they choose. Judge from their reasoning and attitudes around actual transgender/nonbinary people. And in that respect, it's important to remember that bi/pan/multisexual people, on average, have a better track record for accepting transgender and nonbinary people than gays and lesbians do.