I would not. Several years ago, or if the game was initially set up this way, then I wouldn't care. Now in 2020 if SE did this I would see it as virtue signaling. IOW, they wouldn't be doing it because they care about the alphabet community, and I simply don't want the integrity of FFXIV compromised in such a way. Even though some would probably argue it already has been.

Your character has an associated and very clear gender, and the sexual dimorphism in the sexes is blatantly clear in all Eorzean races and their respective tribes. The voice acting in the game and dialogue is done in a way that accommodates whichever one you choose. You need to understand how a change that I am sure you find quite simple greatly impacts the game.

In short, this change is not necessary and it should be perfectly ok for someone to use his and her pronouns.