Honestly not having many scenes where you can see the player and just having text that denotes like the gender could be fine- (Like Jenna's mom saying Young men, could be turned to Young People? Probably better ideas could be had there. I don't wanna tell you how to your game!)
I'm not sure how difficult the software is since I am not a game developer, but even if it ends up being different editions of the game or just a text changer between them might work? As for my ideal world I know to be sort of impossible, so usually I kind of just enjoy the small "Choose your pronouns/sets of parts" for the game to use. That way like, you could play with she/her pronouns and have a dick and you could flavor that however you want(futa, trans, etc...) same goes for he/him with a pussy. Nonbinary could go either way I assume? I don't really know if that could fit into your software though, which I completely understand!
I hope the game development goes well!