11 pages (albeit scanned quickly after the first 3) of mostly unfounded assumptions.
What is the issue with simply being honest and as polite as possible in the given scenario? You have no idea what her DD said to her - maybe she told her mum she wanted a play date at your DD's house and to arrange it. Maybe the woman in question (who's being judged and slated with almost zero facts that merit such reactions) genuinely wants to reach out and get to know you. Maybe she admires your DD and thinks she'd be a good friend for her DD. Maybe she's going through a really difficult time and wants to give herself/her DD some relief and she thinks you/ your DD/your home are safe for her daughter to be at for a few hours. Maybe asking you was so hard but she did it anyway because she loves her DD and wants to be a good mum. Maybe so many things that we'll never know unless she tells you or you ask her. Or maybe, as you assume and imply, she's a weird, socially inept, awkward, snubby CF...
Whatever she is, you still have a choice whether to be negative about her without her being able to defend herself on open social media platform or you could simply be honest, straight and polite
"Hi. Thanks for your message. DD didn't mention a playdate and when I asked her she said she doesn't remember arranging anything with your DD
I was a bit surprised when I received your message - I hadn't realised our DDs were such good friends. And, to be honest, I was a little confused as well. We don't really interact much and I thought I'd annoyed or upset you in some way as I've felt that sometimes you've avoided me. Apologies if I've read the situation wrong.
If our DDs want a playdate, then I'd be happy for them to have one. Let's see how the friendship develops. Seems like the best plan. Let me know what you think"
Easy. If I'd already decided that I don't want this person or her DD at my home, this is perfectly fine and absolutely my choice. I'd simply leave out the final paragraph
And one last but important personal view - from what you say, there is absolutely no bullying or putting on the spot. She's been up front and straight. You always have the choice to say thanks but no thanks. She has nothing to do with your being bullied in the past and you should try to avoid associating that with her (or anyone else not involved in past bullying) and how she's interacting with you. Some may view it as cheeky, rude, presumptuous, etc etc but it is definitely not bullying.