If you claim to fight for equality while also putting down another minority group for their gender identity being valued less than yours on the patriarchical totem pole, you are no less of an indoctrinated hateful bigot than those who support the patriarchy in its entirety. Possibly even worse because you claim moral superiority while having unfounded hatred for another minority who just so happens to be more historically oppressed and less societally accepted than you.
I've seen people claiming they don't acknowledge trans identities because they do not support gender identities in general. That "argument" completely overlooks(or feigns ignorance about) one very simple thing:It isn't about you. I think that a lot of times gender identities are a social construct that only seeks to enforce patriarchical gender roles, but I still respect individual peoples' gender identities because it means something to them and it is easy for me to. To give a similar example: I am atheist and heavily against organized religion however if someone identifies as a christian or other theistic religion and they're not hurting anybody with that religion I don't shout in their face how god isn't real. It makes the other person much happier and it is just so easy to be nice rather than hateful.
I've also seen people claim to be against trans people or gender affirming care being because it can mess up their bodies or people can regret it later. Either they are very ill informed or (more likely) they are hateful and just picked random seemingly sound arguments that fall apart at the slightest scrutiny. Messing up bodies and/or regretting it later is a hazard of any elective surgery. The regret rate of gender affirming surgery is less than 1% while the regret rates of cosmetic surgery are 65% with 83% of the people who had it saying they wouldn't have any type of cosmetic surgery again. It is just a very bad faith baseless argument that shows either major ignorance, major hypocrisy because these same people absolutely will not speak out against other cosmetic surgery, or both.
Ultimately, just be nice and respectful of your fellow human. If they say they are a certain gender identity just take their word for it because it will not affect you in the slightest while it can make their day.