The Crown's gender is completely up to the player's personal headcanons, it won't come up in the story at any point, so if you decide your Crown is trans then they're trans. Choosing your Crown's pronouns is also disconnected from that. If you give your Crown she/her pronouns, for example, the game doesn't automatically assume the Crown is a cis woman; it's purely about how they wish to be addressed by others.
Also, it's important to note that in Arsur, gender isn't assigned at birth. So in the game, being cisgender or transgender doesn't really exist in the same way, if that makes sense. Everyone decides individually as they grow up how they express their gender and what feels most natural to them. Some might physically change their bodies, others might not--you can also headcanon that as you like, since the game won't go into that for the Crown either.
Hope that clarifies things for you!!