My BF has similar tastes with regard to meats, however he does not eat any seafood. We have eaten at Remy twice and he was always able to find enough on the menu for him. One visit they did not have any chicken, but they had pheasant and that was his adventurous meat of the year. He said it tasted like turkey. However, as limited as his taste is with regard to meats, he is quite adventurous with regard to anything else and would eat any vegetable.
My experience with Remy was that I actually liked things that I normally don't enjoy (fennel, scallops) as it is just prepared so well.
I would suggest to go and talk to them at the beginning of the cruise. Get someone explain the menu to you as the written menu is not very descriptive. Our servers always took great care to describe the dishes in detail and from that you would be able to make a more informed decision. As long as you cancel your reservation by the night before there will be no charge.
And finally if you want to just enjoy the atmosphere, you could always go to the dessert experience. No fancy meats involved there!