I too had several of these awful trees. I have two Golden Retrievers who constantly run around the yard and the gumballs were always getting stuck in their paws. I called an Arborist about how I could control them and he suggested that cutting the trees down was the only effective method. I hate killing healthy trees but i also hated seeing my dogs limping around everytime they went into the backyard so I had them cut down.
Then last year i noticed that my neighbor's gumball tree was growing over my fence and causing the same problem. i asked him if he minded having the limbs pruned and that I would gladly cover his expenses. He did one better - he had all of his cut down too.