I need help with gender options in my game

I'm making a video game, which is sort of a mixture of a puzzle game and interactive fiction. I'm a little uncertain about some name and pronoun choices that I currently offer to the player and I thought that you guys might be able to help me.

The game is in English. At the beginning of the game, the player chooses the main character's name and pronoun. This is presented through two screens that offer the choices through textual narrative. It goes something like this, with [brackets] marking the options that the player can currently choose between.

This is the story of...
[...Alice Aster.]
[...Alan Aster.]
[...Al Aster.]

It is...
[...her story.]
[...his story.]
[...their story.]

Detached from the wider narrative context, this method may seem clunky, but I believe it works within the game itself. Mechanically, that is. I'm less sure about the options that I'm offering.

The player can choose any of the three options in the first screen and again any in the second, regardless of what they chose in the first. This affects the player character's name and pronouns used throughout the game.

Now, there clearly are also many other pronouns that people identify with in English, just like there are many other names. However, for technical and design reasons, it would be challenging for me to have the player freely type in their preferred name or pronouns, and neither can I really present a long list of options. At the same time, by condensing all non-binary choices into the most common (?) "their" and by assuming that "her" also equates to "she" and so on, I wonder if I end up coming across as someone who thinks they are on top of things, but clearly has only a very superficial understanding of the topic. Which, to be honest, might not be that far from the truth.

Similarly, of the three names offered, "Al" is intended as a more gender-neutral or non-binary option than the other two. Does that make sense? Would there be a better way to handle this? Are there names that better signal non-binary or gender-neutral identity?

Or am I simply approaching this wrong?

The game itself does not deal with gender identity. As you can see, I'm not the right person to write about the topic. The choice of gender in fact has relatively little effect on the story itself. The player also has no choice over other matters of identity, including their character's cultural background or family structure. The character is not intended to be the player, but someone whose story the player follows. But it still feels important for me and for the story to offer a choice about the name and the pronoun. And I wouldn't be comfortable with it being just a "traditional" choice between male and female, as it would quite explicitly imply and reinforce assumptions about the world that I think we should move away from as a society.

Not that my game is of course going to change the world in any meaningful way. But having worked on it for about six years now, it has been one long personal learning experience for me. And this feels like another opportunity to understand something better.

Thanks in advance for any thoughts and advice.