Actually, it hadn't occurred to me to do so. I was so concerned about not being able to run the fiber inside to get it to my wiring closet, I hadn't thought about just having the ONT remote on the inside of the house. DOH.
So, yes, the cat5e comes out on the outside wall of my garage. Just pull it inside, run the fiber through the wall, and mount the ONT and power supply in the garage. Problem solved.