Oh Heck no!
When kids are in my home or with me out and about, they know the rules, because I make it clear , I do not put up with whining, tantrums and begging.
"We use our manners. We use our words and we respect each other."
And If a child cannot keep it together, they have a choice. A quiet time for a few minutes, or I will call their parents to come and pick them up.
Same for my child. If she behaved poorly while a guest was over, she would have to tell her visitor goodbye and apologize that she was going to be in time out. This never had to happen because she knew, I do not threat, I promise, I will follow through.
This girl being rejected is not your problem. It is her problem and her parents problem.
You need to protect your daughter and you do not deserve to be so stressed out by someones child.
You need to find the words to tell your friend, you are sorry, but you are just not used to the type of behavior you are experiencing with her child.
Tell her you are concerned about her child. You are concerned about your friend, how on earth does she get through each day? I have found that being sympathetic but honest, is the only way to handle these things and then I am willing to let the chips fall where they may.
If this means in the future you only go out with your friend and no children, then that is the way it needs to be .
I have a VERY dear sweet friend that has a daughter that has some issues. We are all so close I am even friends with the Grandparents.
For years we tried to make things work. Once our daughters were in college, My daughter told me, "I am sorry, but I will not be able to be around her any more."
Having to tell my friend and even the grandparents was difficult. I started the conversation and then our daughter handled it very graciously and apologetically, but she was honest. They know this is a problem for this girl. They were not surprised.
Our daughter was so relieved, and so was I.
Tip toeing around this is not doing them any favors.
Take mom for a drink, alone to do this. Give her a hug and tell her you can tell this is wearing on her. And then let her respond. Remember this is not YOUR fault.