I believe that is called a Variable Roll Out Transmission. File a Patent and sell it to GM.
I have the same problem with one of my cars, I now use contact cement on the inside of the rubber, so the insert, foam, and rim are all glued to together.
Another thought (Although I have not tried it yet), is most good bicycle shops, they sell kevlar liners that goes into the tire (Between the tire and the tube), it has 0 stretch, you could clue that to the inside of the tire. It comes in different widths.