If possible, invite several friends of Daughter and Romantic Interest to your home for a game hangout or movie evening, LW! Four or five total. That takes the pressure of them (self-imposed) and if it goes well will assuage any concerns Romantic Interest's mother might have about it.

I personally think it would be better if they hung out at the other kid's house, given this woman's attitude, but if his mother has reservations (justified if she's met LW), she's probably not going to invite anyone.