RE: Playdate...like the mom, but child is rough
I was in a similar situation where I liked the mom, but her boys were WAY too rough with my daughter. She did not handle it well at all, which is different than your situation. Instead, I stayed in touch with her via email and phone, and we went out without kids a few times. I resolved to never meet at her house again for play (because it was too crowded and made everything worse) and to never meet again during school vacation (because it made her temper short with her kids and worsened their behavior). So I would recommend picking your times and places carefully.
The next time we were with her kids was 6-8 months later, and her older two kids acted much better. Her little one was still a big hitter, and my DH and her DH hovered around him all the time. He still hit my DD once and broke one of her toys. So, hover without apology if need be, but don't expect to prevent everything.
On the child side, my daughter liked two neighborhood girls at our last place, but they were kind mean and exclusionary to her. But DD loved them. I did things like inviting them to our house to play, to put DD in a position of sharing HER toys so she could expect and remind them to share THEIR toys at other times. I also taught DD to say "I don't want to play with you if you act like that" or "Friends don't act that way" so she knew what to say when they were being mean. But, most relevant to you, is that I did not encourage the friendship as soon as there were alternative. A nicer neighborhood girl moved in and I encouraged that friendship, and once we moved to a new neighborhood, I never really mentioned the mean girls again, never asked if she missed them, wanted to see them, etc.
My cousin, who has three kids aged 12, told me that parents can and should shape their children's friendships. You can decide who to invite over, who to ask friendly questions about, etc. After our experience with DD liking friends who really weren't that good for her, I have taken that mesage to heart.
Advice and commentary on living overseas
DD1 15, DD2 12, and DS 9