The key thing to remember about mounting any camera is stability. What makes Go Pro so attractive to many cinematographers are the many mounts available for it that fit the camera snugly on a helmet, car, surf board, or what have you. There will always be some motion when attaching a camera to say a helmet, but with the Go Pro helmet mount, you have minimized any residual motion. The only motion that is left is that of the helmet moving.
Trying to mount a camera to a baseball hat and expecting the stability of a helmet mount Go Pro or Contour camera is not easy or trivial.
First of all a baseball cap is not rigid like a helmet, there is flex built into the cap even in the beak. Adding an adapter may make things worse. You will have to experiment. Consider ways to make the cap as rigid as a helmet and ways to attach the camera so it moves with the cap without any swaying or extra movement.
For every design you think will work you will need to make test shots that cover the conditions of the actual production.
I think this can be done. But without your camera or your cap in my hands I would not venture a guess on how you go about it. This is truly hands on experimental kludge-0-matic work.