You poor thing, you sound totally overstretched. Please don’t make any decisions when this sleep deprived! You need to be able to think things through calmly and rationally, and you won’t be able to do that right now.
My DH sleeps in a separate room at the moment, and he frequently did when our older child was small. While you’re asleep is not exactly quality time! If your DH was in a separate room, would you still spend some time with him before going to sleep? What about sleeping apart during the week and together at weekends?
Is your DH doing what he can to allow you to nap at weekends and catch up on a bit of sleep? Have you asked him to? It would be totally reasonable for him to commit to taking your DD out for 2 hours each weekend day so you can get a block of sleep. Then you commit to 2 hours that he can have without interruptions to do whatever he wants to do (and you can’t complain or restrict him about what he does). The weather is warm enough that he could take her out for a walk to a not-too-close park, push her on the swings a bit, and read a book and have a snack on a bench and then come back.
Small practical tip for sheets - put on DDs bed a waterproof, then a normal, then a waterproof then a normal sheet. That way you just have to take off the top two layers when she’s sick and the next ones are ready underneath - saves so much hassle at night. If she’s sick again, just mop up the worst and swap her to the other end of the bed so she’s not sleeping on it....
What are the reasons for not going part time? Is it financial? These decisions are really difficult to make, but can you make it work if you really have to? E.g. reduce mortgage payments? Or is it totally impossible to go part time?
Regarding your daughters cough, what has the doctor tried? Sometimes you really have to keep pushing to get a resolution. If she’s got rubbish on her chest still, even if it wasn’t originally infected I bet it will be by now.