OP I can see why you have been hurt by this. Your dd really never has had a friend. You met someone who was also on their own with a dd who didn't have a friend, and the miracle happened- they both had a friend. I don't think there's any reason to doubt this as genuine, and sounds like both children get a lot out of their meet-ups, sleepovers and so on.
However, it is also obvious that this is rather a one-way street in terms of effort, so you are making all the effort and all the welcome. It's also clear this woman has a narrative of her and her dd being complete victims, and you aren't part of this tale.
It would be a shame for your dd to lose her friend, and I genuinely think the friend would be gutted too (sounds like she likes weekends at yours!) However, I do think you also need to back off the amount of contact, just because this is going to be very difficult for your dd if and when it all collapses, as well it might with the mum being the way she is. Perhaps an occasional sleepover is the way to go. Continue with the teas if it suits you and your dd. But don't mistake this woman for a good friend, as she is not.
In some ways the friendship is working for you and your dd, so don't discount that in your haste to call her on what she wrote. But you can step back a bit emotionally. If your dd has found one like-minded friend, there will be others out there (who also like dolls/games) and the challenge is, I guess, to try to find them and don't make her overly-reliant on this one friend, as they may back off or even disappear, not even for the reason that you have mentioned, they might move or change schools or not be able to come any more.