I doubt that he is only filing for custody on one single issue of missing 1 doctor appointment. What's the rest of the story.
And if you move, yes you have to bear the added cost of transportation.
And why should he pay more? Does he have other kids to also provide for? Has he started earning more, Do you ACTUALLY provide everything the child needs or does dad help out?
Quite frankly the cost to raise a 6 year old is a heck of lot less than to raise a 2 year old, unless there are unusual needs.