Hi CSE.... you are on the right track.
No need for a shorter brim.... the problem is your sessions is right on the brim.
I use a GoPro Tripod mount that raises the cam up just enough.
Most important.....You also need to set your sessions up as "Super View".... that gives a full view of all the action (see pic below)
I will make a new post about my " Kook Hat Cam" with the full super easy set up, so others may find it easily.
The pics you get of action, friends and wildlife are very cool and it is so easy to wear and use.