My husband used to be like this. Until I just started calling him out on it because I couldn’t listen to another day of his bullshit. I would say things like:
“Morning. Ready for another day of dying today? Be a dear and do it quietly, we’re all bored of listening to it”
“Oh dear. Everyone else in this house has just got a bit of a cough, but no, you’ve got the plague.”
“Ill again? Guess you won’t be going to the pub with us tonight then. You can stay here until you’re feeling better”
“As you’re you’ve been so ill lately, I didn’t order any wine, beer or junk food on the shopping order. Those things won’t be helping, will they?”
“Daddy won’t be joining us for our fun day out today kids, he’s feeling ill. Surprise, surprise”
My MIL constantly moans about being ill, I think it’s where he’s got it from. But since I’ve stopped giving him attention about going to the doctors or whatever, he’s stopped doing it. I don’t think he’s mentioned being ill (apart from when he genuinely has been once or twice a year) for 6 years or so now.
I sympathise though OP, it’s relentless and trying to be a decent, concerned partner just fuels the fire.