I don't think *that* argument is one you're going to win....
Without being pessimistic; I don't think any of us would. It's possible he's had gender issues in the past. However it's also quite likely that he hasn't. For me, he sounds like a transgender non-believer. They do exist!

Essentially he just doesn't think transgender, as a thing, exists. Obviously you exist; but you aren't transgender because that's not a thing. It's a delusion.
The same proviso could be applied to an argument about anything: ghosts or crystal healing or coffee shops. Yet it isn't that uncommon a viewpoint, especially when applied to mental health or conditions. Whilst transgender demonstrably exists as a thing, and we know it does more than most, you'll get deniers. There are those who vehemently argue against the existence of autism or schizophrenia or a load of other aspects of the mind.
I really don't know why this happens; people are so weird. But you'd probably have an easier time trying to convert a Bishop to atheism. It's very likely to be some sort of entrenched belief within this man; so he argues against it by the simple counter that all evidence of transgender is spurious because it can have a different explanation. A bit like saying: a light in the sky is just a plane, crop circles are made by people, funny noises are just funny noises, radio static is random etc. As counters to the posit that there are flying saucers.
Have a hug, and see if you can talk to your friend without him being there as she seemed the more accepting of the two. Maybe you could sort things out with her and leave him out for a bit? I know it's such a terrible feeling when you don't get on with a partner or whoever, they're essentially forced to choose between them and you. And it's generally the partner.
It could even be this man doesn't like transgender people in general, so doesn't want his partner (your friend) to associate with you. So he tries to be off putting when you meet him? That's an absolutely typical 'man' behaviour!!! Belittling someone by repeated disagreement.
I know that last bit goes against what I wrote earlier, but they're both possibilities. Either way, how much is the friendship worth and how much pain have you been caused?