Oh OP you sound like such a loving and understanding mum.
My DS does the competitive illness thing. In his case I think it's down to his birth mum only ever giving him attention if he was ill, and telling him that his older sibling was her favourite because they were "very poorly and nearly died" as a baby (which I do not believe but that's another story.)
He genuinely does feel ill much of the time and I've witnessed him vomiting frequently, but generally his physical sense of wellness is completely tied up with his emotional state. If he's having a good day with some activity and some form of interaction, he feels well enough to do most stuff. If he's having a day battling his depression and anxiety then I'll get a list of physical ailments that are bothering him.
He does acknowledge that his mental state informs his physical state but he is not yet ready to get the help he needs with either.
It used to be that if I was ill, he would get very anxious and instead of seeking reassurance that I was just a bit poorly and would be fine soon, he would punish me for being ill. He was pushing me away because he was terrified of losing me. (He has suffered many family bereavements which has intensified this fear.) If I said "Oh man I've got a headache" then he would have a migraine. If I said "Wow my feet are tired" he would tell me that both his legs are incredibly painful and that's why he hasn't done the washing up. If I mentioned I had a bad night's sleep, he would claim he hadn't slept at all for the whole night.
I would sometimes claim to have a really heavy painful period just so he couldn't one-up me 😂
He is much better about managing his anxiety now and if I need to go to bed and rest he will ask me if I need anything and will feed himself and just wait for me to feel better.
Are there any times with your DD (when you're both well) that she is co-operative and is able to talk honestly about her feelings, and how her behaviour affects your feelings? Is there anyone she talks to (at school, CAMHS, other relatives than your mum) and seems to trust who might be able to help with... I guess "mediating" between the two of you to help you both agree to changing your behaviour on the stressful times when you are ill? (No you shouldn't "have" to change your behaviour, but the important thing is that she sees you are willing to compromise in order to meet her emotional needs.)
In purely practical terms, could you have a day, maybe with both DDs helping, of batch cooking some meals that could then be frozen and used if you are too unwell to cook? Then all DD1 would need to do would be to sling something in the microwave and not have to deal with anxiety about cooking on top of her anxiety about you being ill. (Would also be useful for you if you're knackered one night and dread cooking!)
Chilli is a great one for this, especially if you keep some microwave rice packets in the store cupboard. So just cook the main part and freeze that. I do the same thing with spag bol - cook up the meat sauce and freeze that, then all anyone has to do is microwave it and cook pasta. I do several types of soup like that as well, and things like chicken stir fry (choose your vegetables wisely, as many don't reheat well. Looking at you, red peppers!) and chicken curry (again just cook the meat and sauce and use with a packet rice.)
I hope this helps in some way OP. Again, you sound like a wonderful mum :)