YANBU ... I understand how it's a shame that this mum hasn't taken the day off to be with friends, but for the sake of her daughter, and the AP please be willing to take them both.
Maybe a compromise would be to contact the particular mum - and not by e-mail but on the phone. Ask her to ask the AP if she wants to come with you - the other option, a day off (which I am sure would be welcome - she could maybe go shopping, meet English or Polish friends, or just have a day to relax other than any other day(s) off she gets).
Yes, the mum was undiplomatic in assuming it would be OK to 'send' her daughter (and taking the mickey when it will mean someone having to go out of their way to give both a lift each way), but we don't know why she is unable to come, nor whether she was stressed from work when she wrote the e-mail...
If you unilaterally exclude the AP she's either going to feel miserable, or overjoyed, but it would be nice to allow her to make the decision so if she wants to be there for the daughter, she can, while if she'd prefer a break, it lets her choose that. It certainly treats her as a person, with an opinion, rather than having been told one thing, and then be told something else. She may understand that it is a bunch of friends, and gives her the chance to try to 'fit in' but doesn't pressure her, or exclude her against her will.
If she wants to come along, then although she's not the person you'd prefer to have been there, she will hopefully enjoy the day out as a complete change from the rest of her time.
I've only got to the end of the second page of comments but seeing suggestions of 'take only the daughter' made me feel for the AP!