I agree it’s weird that your mom is being so persistent. I also agree they she needn’t be included in “the” birthday dinner? But it’s a whole birthday weekend? Does your mom just want to be included for a little bit, but not the whole weekend? I’m reading between the lines here.
Your bf wants do just dinners wish his fam but he’s ok with your parents “hanging out”? I think that’s more awkward though I guess they could do a short visit for afternoon tea or something. I also think 45 min away is local enough depending on your area.
Maybe you could broach it as mentioned above like your parents as locals want to invite his parents for lunch? All depending on their schedule.
I totally get it’s awkward with in laws. I’m not clear if your bf even brought it up with his parents. I understand maybe he doesn’t want to do a dinner with everyone but as he suggested something short so people can start to know each other. So bring it up again and follow up on the “hanging out”. And then tell your mom what your decision is.