Disney Cruise Line Blog has all the menus for the restaurants on board (they are updated frequently and dated; if it looks like an older menu that just means they haven't changed the menu since that one was posted). You can take a look there at what's offered and hopefully find something that will fit your tastes. And, as others noted, they will accommodate your tastes.
Though I will say, if you find you would be significantly altering menu items, then Remy might not be worth it. It's expensive, partially because of the level of the dishes they serve, and if you are 'taking away' a lot of that it might not be worth the price. I'm certainly not telling you what to do, just noting that if you find you are significantly altering the dishes then maybe Remy isn't the place for you. But if you're just asking avoiding certain dishes or asking for mushrooms to be left out, that's pretty minor.