Compressing the circumference of the tire with a ratchet strap or two with the valve core out and then putting a few seconds of air at 90+ PSI into the tire should seat the beads. Note to not exceed the bead seating pressure seen on the sidewall, generally 35 PSI. I slowly collected a bunch of several inch long honey locust thorns embedded into in the front tires of my 75 HP utility tractor cutting overgrown fields and demounted the tires to pull them out. I successfully used this method with two 1000 pound straps to reseat the beads on those stiff 8 ply tires. Getting the tire back on the rim was much more "fun" than getting the beads seated due to the stiffness of the sidewalls.
Inflating the fronts to the maximum allowable pressure certainly helps with keeping the tires on the rims when using a loader. However, having enough ballast on the 3 point helps even more. The last time I saw somebody roll a tire off of a rim on a tractor was one with a known small leak in the front tire and nothing on the 3 point. They picked up a bucket full of mud and pop went the bead. On the plus side the leak got fixed, they'd managed to get a small piece of a stick wedged in the bead seat to cause the leak, and the stick was easy to remove with the bead broken!