I think your mom’s persistence is very odd… she should be able to handle not being included in her kid’s boyfriend’s mother’s birthday dinner/weekend plans. Have you asked her why she is so invested in this? It’s neither weird nor wrong for his family to come into town without having some kind of get together that includes both families. You’re not obligated to invite them to hang out at any point during the weekend.
I also don’t think it’s weird at all for both families to get together for a meal even though they don’t know each other well — that’s how they’d gradually get to know one another, if that’s important — but since both you and your boyfriend are uncomfortable with inviting your parents to join this specific weekend, don’t do it. I’ve been with my bf for over five years and our families have barely interacted. My dad and sister have met his mom, stepdad, and bro/bro’s family exactly once.