An aired down radial has way more sidewall flex than a bias ply. My old 36" TSL's didn't show any sidewall bulge even as low as 7 psi, and my new 40's are the same. A radial at 7 psi is considered flat, or damn close to it.
As for thick sidewalls...not really. For thick sidewalls you need to go to Michellin's. I've seen a TSL cut up, and the sidewall is no more than 1/4"-3/8" thick. The Michellin was easily twice as thick.
The bias is tougher than the radial, but it's also rougher riding, has more rolling resistance, and will be wore out that much quicker.
Rene