"I have known dp and his children for many, many years and she was exactly the same when she wasn't a tween. Yes, she has no social skills and comes across as extremely rude, but she is also cursed with a terrible superiority complex. She is unkind to people and ridicules them. She doesn't have 'strops' but she is quietly manipulative and can turn the mood of a day so that everyone has a rotten time.
She just isn't very nice. Why is it OK to say that about an adult, but if you say it about a child it's DREADFUL? Some people aren't very nice."
Because you are supposedly an adult and supposedly above it all and able to step back and stop taking it so personally and not object to every fibre of a child's being. She's not "people", she's a child*. You don't judge a child by the same standards you would apply to a colleague or another adult. You are coming across as incredibly immature. That is what is dreadful.
'I can live with her not being very nice, the issue is her father and how he handles it, isn't it? He goes mental at the idea that I don't want to spend every other weekend with him and his kids, rather than just calmly being able to talk about it. He makes excuses for her and says she is absolutely fine with 'other people'. My family and friends have all met her and found her the most strange and rude child they have ever met (my parents were in teaching for decades) They feel sorry for me for being in this situation, I am not regarded as a terrible ogre. PS I don't call her 'awful' to him. Or strange.'
OP, there is no hope for this relationship. The issue is that you and this man have a completely different idea of how to handle this child. I suspect you have burned your boats already with him where his DD is concerned. You can't accept his view of his DD. He won't back down and listen to your pov. You really do want him to choose between his DCs and you if you are campaigning for child free weekends with him. That is simply not going to happen, and really, how dare you even suggest it, that he should give up his weekends with them essentially, or go elsewhere with them when he has them for the weekend.
What I see in your reference to your friends and relatives sharing your pov and finding this child strange and rude is a really ugly picture. Tell us you haven't gone whining to a large group of people and tried to turn them against this child, or gossiped with them about her behind her (much smaller than your) back. Shameful.
You are showing yourself to be really pretty manipulative, stroppy and not very nice yourself where this child is concerned, not to mention cold and unkind and cursed with a superiority complex, imo.