Foam filling might solve your roll off issue. I say this because I recently had one of the foam filled tires on one of my company's rough terrain fork trucks roll off the rim just like what you describe. The reason I bring it up is that the tire had a large gash in the side but stayed on the rim for quite a while (months) until enough foam broke loose and fell out. Once enough foam came out the tire lost its rigidity and was able to work its way off the rim. Adding foam would have the opposite effect, stiffening the tire and making it harder to work it's way off.