I have two teenage DSCs and they are here all of the time. My DSS is 15.
Unlike the OP, my DSCs live with us as a normal matter of course. But in the situation the Op describes, the child would not normally be at the home on the days in question, so yes, of course the stepmum should be consulted...I don't care for the phrasing of "asking permission", but rather, the decision to alter the usual arrangements and how to manage that change should have been a joint one between both the adults living in the home.
The idea that at 15 the child no longer requires 'babysitting' - well. Yes and no. Many 15 year olds do require supervision. In fact, I was recently told off here for leaving my own 15 year-old DSS with only the supervision of his 18-year-old sister, and basically told that if he trashed my house while I wasn't here, it was my own fault for leaving a 15-year-old alone because 'what did I expect'.
If it isn't acceptable for the girl to stay in her mum's house while mum goes caravanning (and I agree that it is not), then why is it acceptable for her to stay supposedly unsupervised at her dad's house? Obviously, all the adults in her life feel some degree of supervision is warranted, so it is spurious to suggest that the OP can just act as though nothing additional is being required of her under this change in plans.
Having teenagers around all day in the summer, especially when they are not inclined to do anything, is bad enough. Having no authority to enforce house rules is unacceptable.
OP, can you and your DH agree on what the house rules are? And how they'll be enforced? Because reading your OP and especially your subsequent posts, it seems to me that the real problem is not these extra days with your DSD - but the fact that everyday life with your DSD is not good. She's allowed to come to the home and act as she pleases with no responsibilities and you are afraid to call her up on things.
Forget about the chilliness and the lack of conversation...that's beyond your ability to address, I think. What you can insist on is that house rules are followed, and that she contribute to the household in appropriate ways, and that she doesn't make things difficult - practically, I mean, not emotionally - for you or others in the house. At 15, she can make her own meals and clean up afterwards. As a minor in your home, she can cheerfully comply with reasonable requests from you as the Lady of the House, like "could you put the hoover around, please?", or "after this programme, I'm going to kick you off the telly; there's a show I've been planning to watch", or (one of my bugbears!), "could you clear off and give us a bit of privacy now please; my friend and I are just going to have a bit of a chat. thanks, pet."
You have to feel that you can come and go - or not - as you please, and you can't let any teenager (your own or not) control your mood and your actions, or you will end up drowning in a sea of depression and anxiety.