Ok, you are facing 2 problems there:
If the tire is a rubber than the tire walls are way too thick to distend/collapse them nicely.
Try cutting/scraping the rubber away from the inside till you only got a very thin skin left, this should do the trick. Turning the Tire inside out will help.
Adding a bit of weight into your model will also help, as well as drilling a hole into the rim for air to escape
If the Tire is an injection molded tire than your best deal is to chop it and use putty to redo the widened part of the Tire.
Just some thoughts from the top of my head.