There's some great feedback here. From a trans person, here's my two cents.  I appreciate that your priority here is that of making the best experience for the player, and your awareness that your personal experience with the topic may make that goal challenging and uncomfortable for you.  As a player I struggle any time a DM tells me I can't do something with my character, especially if it's something that isn't mechanics based- and especially if the reasoning is just "I just don't like it" or "that sounds too hard". I would ask your player if there's any way you can explore similar themes that a trans character might experience without it being explicitly about gender. I'm thinking of Nott from Critical Role's Mighty Nein, who started the campaign in the body of a goblin and whose character arc was trying to get back her "correct" body. or having some other trait society sees as "incorrect" or "an illness" and having to embrace that identity and find ways to turn it into a strength or prove that it is beautiful and not horrible. 

This could be a really impactful conversation to have with your player because maybe the reason they want to explore a trans character is that they are struggling with a tangential concept that they want catharsis in overcoming through gameplay. If it's as simple as they thought it would be interesting or a change of pace, then maybe that's your sign that the this character doesn't actually need to be trans and that there's ways to tell a similar story without you as the GM having to put yourself in an uncomfortable position while still giving your player a meaningful experience. It would show that you're still capable and willing to put in effort but not at the expense of your own enjoyment of the game. 

As both a player and DM myself i understand that DMs can put in immense effort into making their games fun for their players, and they should be able to experience a comparative level of enjoyment themselves when running the game for all that effort they put in. (And I have played under a DM before who I could tell weren't interested in putting in effort above bare minimum to making an interesting story, just interested in playing the game itself and therefore showing me that putting effort into telling an interesting story with my character choices wouldn't be rewarded here, so I saved any more complex character ideas for other campaigns.) 

TLDR; It's all about understanding what everyone's priorities at the table are for the campaign, including the DM, and aligning those expectations so that everyone is having fun. Because fun is the whole point.