You realize that there is no evidence for either the advanced medicine or the magic required to take a person who was born anatomically X and make them anatomically Y in the setting, right? Certainly not post-Age of Legends, anyways...
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And, no, the act of an entity which for all practical purposes is a deity does not count as evidence that anything else in the setting can do that - particularly given that the 'transformation' is literally reincarnation.
So if you're looking for a reason why trans people in the sense of people who were born anatomically X but have become anatomically Y don't exist within the setting, the obvious answer is that the capability to perform that transformation doesn't exist within the setting, at the very least not in any time vaguely close to the 'present day' except in the crude form of castration.
Also, I would suggest that the true solution to this 'problem' is to change how you think, because your description of the 'problem' is basically a "boys have penises and girls have vaginas so trans cannot exist" argument. Failing that, the best 'solution' would be to acknowledge the "male souls / female souls" thing to be how the people of the setting think things work and just leave it at that, much like how many Star Wars fans regard Qui-Gon Jinn's statement on the relationship between midi-chlorians to be a mixture of fact (high midi-chlorian counts corelate with strength in the Force) and belief (midi-chlorians are the interlocutors for the will of the Force). Altering the setting's metaphysics will only create problems, especially when your rationale for altering the metaphysics basically boils down to you reading it as boys having penises and girls having vaginas means that trans cannot exist.