Many women, and this comes from a woman friend of mine, go more for the fantasy than the reality. Pushing a woman into a ‘extra-relationship’ goes against her instinct of finding one ‘mate’. They feel you don’t really care for nor respect them. Sounds like this is the case of your girlfriend. So I would talk open and honestly with her of her thoughts and needs of a relationship, and if you are not totally on the same page as being a couple, I would break up and let her go her own way and you find a woman that will be your ideal woman. Otherwise, this scenario you laid out sounds like it eventually crash and burn.