I guess that's something you need to discuss with your partner as far as what might be helpful and work for both of you so you don't trigger one another. As far as how to talk to them, I guess could you approach it from a, you are glad they care and want to be supportive standpoint? Maybe you could discuss some ways that both of you get your needs met? To be honest though, eating disorders can really impact your relationships with others, and I guess the thing is when you aren't in a relationship it's easy to ignore the impact it can have on others, but when you are in one, it's often pretty obvious. Even the roommates I've lived with in the past noticed and made comments a lot.
A few things we did on and off that I guess you could consider... but they obviously don't replace professional help or support if that's an option-
-having them eat as regularly and normally as possible so that I can try to either mirror them or at least see what normal eating is like
-them being open to answering questions about their eating habits (if they are normal ONLY- how they know what to eat, when to eat, etc.)
-not having them pressure me to eat/not eat/comment on what I am eating when I do eat (unless i ask for reassurance that what i am eating is okay/allowed/etc.)
-sticker charts for having meals with rewards
-removing scales for both weighing yourself and cooking/food
-removing or crossing off calorie/nutrition labels if that is an issue
-them asking if i want to go grocery shopping together, want them to grab me things when they go, or help with accountability to maybe place an order online if that's the way you do your food shops
-them asking if i would like to be included when they order food out/go out to eat and if i want help with making choices (not just assuming I want to be included or need/want help with what to get)
-again not commenting on what i eat/don't eat/appearance (unless things get to a medical concern standpoint)
-support and having someone to go to appointments with to get professional help if possible/wanted
-reminders to eat if they notice you haven't for a while (and what is okay to say), like, hey i don't think you've eaten today don't forget to do that and move on versus you need to eat and you need to eat xyz now
I dunno it's hard because it's good they want to support you but it sounds like if you are really struggling is getting some outside support an option as well?