My question involves a child custody case from the State of: Missouri

I have a tricky question.

My fiancee has a seven-year old boy who currently lives with his father. He has lived with the father since he was four. They were divorced shortly after his birth and she raised him by herself until he was three. At that time, she was working 60-70 hours a week and had no one to rely on for help in taking care of him, although she was doing a wonderful job considering the circumstances. There came a point where she realized she was either going to have to quit her job, in which case she would have no income, or try to get a different job, which would have likely put her in a bad situation financially as her mortgage was very high. The father wanted badly to have the child live with him. Mother realized she was either going to have to put the child in day care for long periods of time to be raised by strangers, or live with is father. So she made the painful decision of letting the child move in with the father. They never went to court to establish visitation rights. She saw him on the weekends whenever she could. The father lives about 46 minutes away from our city in a small town in Missouri. The child goes to a school that is ranked about 450 out of 500 schools in Missouri, with one of the worst academic rankings in the state. Missouri is already ranked 22nd or 23rd in education, so this is one of the worst schools in one of the worst states for education. We are very worried about his academic future. In addition, the father is unmarried but lives with a girlfriend of several years, they both work full time, the father during the day and his girlfriend most nights. The child spends most of his time outside of school at a day care run by his grandmother, the father's mother. By the child's own admission, he rarely gets to spend time with his father.
Fast forward four years, the child is now seven. I have been accepted to a very good graduate program in San Francisco. My fiancee is going with me, as we have been living together for the past three years. We are leaving at the end of this summer. She plans on going to school next year but will work part-time until then.
We would like to have the child during the school year and have him live with his father in the summers. We would also agree to having him in the summer and let him live with his father during the school year. I am financially well off, as I have recently received a large inheritance, and so I have agreed to pay for the child's transportation to California and back to Missouri as often as his father wishes (winter vacation, possibly spring break, and summer). We are researching schools in and near San Francisco to look for the best option for the child. We can also afford to rent a house or apartment with a bedroom for the child. We care about him very much and he gets along with us very well, although he is unsure about moving. He has never been out of Missouri, and he is understandably wary about moving to an unfamiliar place where he doesn't know anyone. We want to make him as happy as possible while still making sure he is doing well in school. He doesn't have many friends, only a girl at the daycare who is much younger than him. He doesn't seem to get along with the boys at his school, probably due to his sensitive, reserved nature. He is a very, very sweet kid and he is always happy to help with chores, and is very kind and thoughtful, always thinking of others, which is somewhat rare in children at that age.
The father was initially somewhat receptive to the idea of us having him during the school year, but has recently had a change of heart and has told us he is "not letting us take him anywhere." We are primarily concerned with the child's education and in giving him opportunities he would not have living in a tiny town in rural Missouri, although it will be somewhat of a culture shock, I firmly believe in mill be good for him in terms of a high quality education in a city that offers so much to learn about and discover. There are museums, an aquarium, wonderful libraries, cultural richness and wonderful architecture as well as natural beauty in the surrounding region of Northern California. His mother is willing to stay at home for at least the first year and I will not be working while in graduate school, so we will both have the opportunity to spend a lot of time with him.
So, should we begin with mediation, or just go to court immediately? We only have a few months to work this out. As they never went to court to establish custody rights, what rights does the mother have in this situation? Considering the situation by which the child went to live with his father, which was the opposite of the mother being uncaring but rather trying to do what was best for the child despite the painful act of letting the child live with his father raise him for the past three to four years, do you think that will have any bearing on the decision? Can we demand that he live with us at least three months out of the year, if not the entire school year?