One thing I think that needs to be unpacked here is that the Wheel of Time has some very specific opinions with regard to its overarching metaphysical structure and these are very important.

First, the Wheel of Time is an explicitly dualistic universe. The mind is not simply an emergent creation of physical processes but is a separate entity that can exist entirely without a body at all and is instead linked to an immortal existence that cyclically reincarnates through different bodies over the course of time. This is generally referred to as the 'soul' in everyday parlance.

Second, souls in WoT are not infinitely malleable and in fact come with a whole pile of traits. In a sense souls have inherent traits (nature) in addition to being shaped by outside circumstances (nurture). This is made very clear through the Portal Stone journey in The Great Hunt when Rand lives a hundred lives in succession but is still very clearly fundamentally the same person throughout them all, and the recurring line 'I have won again Lews Therin' calls this out explicitly. Another character later comments on his own experience during this event and provides confirmation.

Gender appears to be one of the traits inherently bound to souls. When characters live many lives over and over they always have the same gender. This is true during the portal stone journey, it's true for the Heroes of the Horn, it's true for Rand himself, and so on. Aran'gar, a being essentially created by the Dark One perverting the nature of reincarnation in-universe, appears to be the only exception (there's an unanswerable question attached to what would have happened if Aran'gar had been reincarnated again since she was killed using balefire).

Given all this, the only real way to have transpersons in the WoT is if souls are placed into the wrong bodies. There's nothing wrong with that philosophically, however, no such characters are recorded in the series. Now, that doesn't mean they didn't exist during the Age of Legends since we don't see enough of that time period and the magitech utopia of that period was probably capable of rendering a trans individual capable of passing perfectly so there is no real way to observe such a character in the first place (after all, they could create human replicate bodies in the form of gholam). In fact, it would be possible to have one of the Forsaken explain this.

If such characters exist in the Third Age, well, one of the reasons we are unlikely to encounter them is they probably face immense prejudice. There's very strong evidence for powerful in-universe prejudice against being publicly trans (for one, Min faces considerable prejudice just for wearing some male articles of clothing, even though she presents as female the whole time). Male souls placed into female bodies who channel saidin (the WoT version of transmen) are likely to be particularly vulnerable - plausibly the Red Ajah kills them all outright and then lies and says they had male flesh from the start in the records (this would actually explain why none of the Aes Sedai in Salidar recognize the Aran'gar threat, since none of them are Red). Female souls in male bodies who channel saidar is a less lethal situation, and it is possible that a small number of them are found among the Aes Sedai or other female channeler societies. There is a question of how detectable this would be. The Mirror of Mists can be used to create an alternative appearance and it strikes me as plausible that in addition to whatever surgical means could be used safely, transwoman channelers would present themselves as how they believed they should be using this art more or less constantly.

I think the adaptation could absolutely produce a couple of characters of this nature - probably mid-tier Aes Sedai in Salidar would be the best choice, perhaps even one of the Sitters - with trans actresses, but there would be very little reason for anyone in-universe to ever comment on it.