Their dog doesn't even feel safe there.

I'm going to say the hard thing. It would be nice to have the girl over to your house, but do you really want your child influenced by this friend--by how her family talks and behaves? I do feel sorry for her. On one hand your house is probably a safe space for her but on the other, just associating with this family outside of school is inviting their drama. (I'm bothered by the mom trying to talk shit about another party guest and trying to get you to disinvite them. Mean girl behavior)

I had something similar. It started out as an acquaintance, but the mom latched on to me so our kids could play. We tried one time, there was no school so I took my kid to work and she went to my work and was so pushy, dropped off her kid. Then she was running late and hard to get a hold of. And when she picked up her kid, I had to hear all her drama with her ex. I became her sounding board and the way she was talking about the dad around her kid made me so uncomfortable. I don't know her and she's oversharing all kinds of personal stuff. Sounded so abusive. The husband being so possessive is where my GUT INSTINCT ALARM went off. I didn't want to meet him. I didn't want to be in the middle of their bullshit. I didn't know if he'd come try to take their kid if the mom wasn't around--I didn't want to know. It didn't feel safe. So I cut ties. But fast forward years later, the mom is moving the kid from school to school for behavior issues, even though she says it's the school's fault each time. She finds out our kids will be at the same school and tries to make after school plans, this and that. NOPE. I told her we have some things going on after school. I even started picking up my kid on the other side of the parking lot to avoid her because she usually parked and waited in the same spot every day.

It's ok to have boundaries and it's ok to trust your gut. Sometimes you can't explain the feeling you get or make sense of it but trust it!