I don't think a Bighorn is what you want for a run-flat tire. Mine with 0PSI is flat! On the other hand, my Radial Outlaws could be run with no air pressure. Similarly, a set of Outlaw2 (not the OG Outlaws) seems to hold the weight of a SxS with little to no air pressure. This isn't to say that the Outlaws wouldn't roll a bead being run like that, but I know I have run my Radial Outlaws with 0PSI for a few miles. I just didn't make any corners hard and fast.
Those are heavy tires, heavier than your 589s. I'm not sure what tire you might consider that meets your criteria.