@Hard Times,
Don't get her pregnant. FFS, if you're lukewarm about her then a baby is not going to magically fix all of that.
And don't have kids just because you're feeling pressure to.
The kid/no kid thing should be a 100% deal breaker, because there is no possible compromise. You can't just hand over a child to her 24/7 because you decided, hey, you didn't really want children.
And there is nothing wrong with being child-free! I am and so are other people on this site. It doesn't make you selfish or less of a humanitarian or nasty or child-hating or anything of the sort. You are allowed to feel this way.
But this woman (God, I am 100% sick of people referring to grown women as girls. She's not 12) does want children. For sake of argument, I am going to say that she is committed to having them -- despite what she told you. I think she's trying to make things work.
But just like someone not wanting kids having one and not getting a compromise, the same is true when someone wants children and they don't have any.
You will not magically turn into the first couple, ever, to find a compromise for this. Because one does not exist.
As for the rest of it -- she's running hot and cold for a reason. She likes you, too. But this is the deal breaker.
Be kind -- both of you -- and end it as friends. Oh and by the way, you can still have a lovely time together and have it never become serious. But recognize that if that's the case that demanding exclusivity is wrong and it also means if she finds someone she likes more who wants to have kids with her, then it's over.
And the same is true in reverse. If you find someone you like more than her who doesn't give you agita about children, then you'll be the one to call it quits.