I'm going to have him write up a birthday party invitation to give to her the next time he sees her. That seems a little more concrete and easier for a parent to accept than an open-ended invitation.
I agree that I would be a lot more likely to bring my child to an "unconnected" house for a birthday party than for a random play date, the reason being that I can expect that there will be other kids around at a birthday party. I might even see a parent I know and find the connection I don't have yet.
You might also want to make it clear in the invitation that the parents are welcome to stay for the party if they want, although they do not have to.
