I know you're not explicitly blaming her, but what you're hiding from is the situation your DP has created and continues to allow, not the child.
There's nothing wrong with taking time for yourself. I realised early on to override my fears about not being an amazing stepmum - always on form, making an effort, being super happy fun, planning stuff - and realise that I'm a generally nicer happier person when I get time out to read, nap, chat to other people on the phone, cook something (ALONE). "I will plate your hair but we'll do it this afternoon. I'm going for a walk now. Have fun with Daddy" "The soft play centre sounds lovely and you know how much fun you both have when you 3 go together. I'll make dinner while you're out and see you later". They never feel hurt, rejected or upset, they get time with their Dad, he gets time with them, I get a break.
DH completely supports this and it works both ways. I can never understand these men who insist their DPs are there for every minute of contact time, it's primarily for them to have time with their Dad, he should be taking the lead.
Sometimes he's shattered and I do something with them for an hour while he recharges. Sometimes I leave him and them to it and take myself off for a walk.
We have them quite a lot so it's a normal family life in our house, no Disneying, usual stuff goes on like house jobs, food shopping, lazy Sundays where we all relax and no one gets entertained. Has your DP not realised that now she's with you more than she's not he's got to step up and do some proper day-to-day parenting?
With a recent alteration in arrangements your DSD has had a lot of change and is just acting out. Having time to yourself isn't rejecting her, it's giving her time with her Dad and you time to yourself. Try and separate the issues out. She's not purposely making your life harder, honestly she's not. She may be testing boundaries to see what you'll put up with, but this is different.
My DSS gave us a horrific few weeks when he was about that age. His mum was moving house and the whole thing was a mess, they got pulled out of school at short notice, there was all sorts going on her end. And he acted out when he was with us, because it was a safe space.
I remember one afternoon when he screamed and hit and through stuff for 5 hours. I took DSD out for a walk and to see some friends to get us away from the shouting and DH did a combination of holding him, talking to him, listening, taking away anything breakable, getting hit, leaving him alone for periods, just being there while the poor thing raged. He couldn't have explained what he was so angry and upset about, he was 6, but he knew that DH still loved him no matter how awful the storm was and after the really big outpouring it never happened again.
My point is, that she may be unsettled about the change in contact, as well as your pregnancy, and the combination of following you around and needing your company and closeness, and the tantrums and boundary pushing could be signs of unsettledness generally. This is for her Dad to sort out and help her with and you can try and support him.