Wow! In digging holes, we have discovered huge tunnels 4' deep, and I understand they can even go deeper. BUT we don't have clay, so an easy dig...........until you hit a boulder!! I think cutting the bottom would work fine! We have voles, so we buy gopher wire by the roll, and circle the trunks with 1' high protection. Keeps rabbits away too. You could try placing some gopher wire in the bottom of the hole where you put your bottomless basket. That would keep any adventurous gophers away from the heart of your trees, giving them time to recover from the root pruning attackers.
Way down on our hill, I've planted Castor seeds, and 5 plants are coming up. Gophers will die if they try to taste a root of those, but they know better. The plan is to use a probe around every important tree on this property, find a tunnel, and drop a few seeds for gophers to have for dinner. I thought of planting spurge around each tree's dripline. Gophers don't like spurge. OH also! drop ammonia soaked rags in the entrances of tunnels. They will vacate that tunnel, and move. So if you do that, you gotta keep it up.
Good luck!
Suzi