Look, there's no need to accept this from your daughter or your partner, frankly.
Who's the boss? You're the boss!
Bang your fist on the table with your partner and tell him to take care of HIS children at least when he's at home.
And as to your daughter, tell her mummy is going to do the household. That she can play alone or watch or participate. Whatever she wants. Start by giving her something she really likes doing like the pretend play (which is draining for adults). Either way, if she tells you what the dolls have to say, she already plays by herself, but she somehow doesn't have the self-confidence to really play alone. It only takes for her to realise she can do it to really do it.
If she does start following you around, I'd tell her clearly mummy has not got the time to play and that there are other things mummy needs to do. If the worst comes to the worst, give her her favourite toy, close the door and wait.
If crying does become really hysterical and not theatrical, you can decide to go in again and do withdrawal or timed play with a fancy egg timer.
I do play with my daughter and son, but the default thing is that they play alone. I can leave them (1.5y and 3 months) in the next room while I clear out the dishwasher. On the rare occasions I do hear anything (not hysterics), I call out so they know I'm there. I have always done this precisely because I didn't want to have children who needed to be with me/see me all the time. I need my mental space.
When she was 1, my daughter would sit happily playing in my ob's waiting room while I went to the lo for urine test, and had my weight and BP done or blood taken.
Don't take it as being smug either, it's a matter of showing who's boss. You can't be a good mum if you're drained like this. And it shows a sense of work ethic/duty that we have things that take priority over fun. I think that's a valuable lesson to instil in children.