You could build the run totally around the coop so the coop is freestanding in the middle. Then it could be any height you want and the chickens could use the space under the coop for shelter/shade.
Roller bars are used to keep dogs/coyotes from climbing over a fence, but I'm not sure if that's something that will work for mongoose. Maybe a strand of hotwire? I would think mongoose could climb a fence though. Maybe ask on your state thread what others do in Hawaii to protect against...