It sounds like SD might benefit from being occupied this summer.. since you are working from home.. are there day camps in the area that she and the other kids might enjoy?  Are there things her other friends are invoilved in?  

Ultimately.. if you are working.. the kids should be able to be out of your hair.. 

Stick the Skid in summer camp and save yourself the grief of the presence of a moody 13yo minion of a HCBM.

End of drama. At least during day camp hours.  Make sure that camp hours cover your entire work day and let daddy drop off and pick up his child.

Focus on living your best life.  That may not include being your partner's  and the HCBM's default child care for their failed family progeny.

If the SKid is polite, kind, and engaging, then engage. If they are as  you describe your SKid, dump them in daycare/summer camp.  Save yourself the drama and focus on your far more peaceful work day sans the failed family spawn.

I would.  Not your spawn, not your problem.  Being our best for ourselves, our partner, and even our SKids may take active mitigation techniques for the baggage many SParents are saddled with and can often require focus on mitigating the problem which can mean eliminating the presence of the drama delivering failed family baggage  as much as possible.  Well parented reasonably behaved SKids are a very different thing to deal with than are noxious progeny influenced by a HCX.

DH needs to have a set plan with SKs for the summer. Has he enrolled SKs in camp? Is he taking off work? What will schedule with BM be? Make sure the SKs are not dumped on you this summer!

Try to find a  '''camp''' SD is interested in. Creating a web page.  In and out of tik tok. 

Been there, done that. I endured my SD35 shunning me while living full time with us when she was 15 to 22. DH was a chicken shit and did nothing but put the onus on me because I was "the adult." I had gotten my coaching credentials at the time and was going through my own coaching and counselling and was taught to make three attentps a day to engage with SD. That bitch was relentless. She'd answer but continued to shun me. It went on for seven years. I almost left DH because I couldn't stand it and he never had my back. He'd gaslight me and say that he can't make her life me so I was the one who had to eat shit.

Well, we went to therapy because I was about to leave. The therapist told me to never make attempts for contact with SD again. That only gives her all the power to keep rejecting me and it reinforces the message to me that I'm a big zero in the family and that no one has my back. Therapist said that SD is well aware and the longer this dynamic goes on the more power she gains. Therapist said do not do another damn thing for SD. No rides, no contact and that DH must have my back. That one made DH squirm. Ultimately both DH and the thearpist said word for word, "if she gives you snot give her snot right back."

So, bottom line is do nothing at all for your SD. Do not pick her up from school, don't give her any rides at all. She's no longer allowed in your vehicle. Do not do any favours for her and do not engage. Do not ask her how her day is or what she did in school. Nothing. She's 13 years old so she shouldn't need you to look after her. Stay away from her when she's in the house even when your DH isn't home. During summer breaks, I would make sure she's not in my space at all during the day. I wouldn't want that dark nasty energy in my home. She can stay at her nasty BM's, who probably trained her to be the bitch that she is or she can go to another relative's house or some kind of camp. But she cannot be in your home when your DH isn't home. I ulttimately told my DH that. My SD acted very sucky and like she needed care because she was crammed so far up both her parents' asses, but I told DH that it's not my problem. If I'm on vacation days from work, SD cannot be in our home unless DH is there. I will not tolerate that nasty energy. SS was very engaging with me, so I was able to continue to remain engaged with him and give the perception of favouring him over SD. I didn't give a damn. In fact the therapist and even DH felt that it would send a message to SD that there's a difference in how I treat SS compared to her as he treats me way better than she does.

So, yeah. Disgenage to the point that you shun the little bitch right back. I mean it's what she wants. It was actually very interesting seeing my SD squirm when I started shunning her right back. I think these nasty girls expect to always hold the power and when they realize they've lost it they react. Start your shunning today and grab your popcorn. It's also very cathartic when you realize you've taken your power back.