I'm assuming the suckers are all the palo verde trunks in the picture. That looks like an awful lot of tree and an awfully big root to try to move. If you had a smaller sucker, not more than a few feet high with a much smaller root I think you could sever the root between the sucker and the main tree, then take as much of the root going the other way and move it and probably have pretty fair luck. Also probably best to make this move in the fall and give it a bit extra water to help the roots get established before the next summer. It seems to me that I have seen these trees sucker where roots are very near or above the surface and maybe suffer some damage such as some cut bark or such that stimulates new growth.

Your palo verdes should be getting absolutely gorgeous in a few weeks. Wish I could get down there over the next few weeks when they are blooming.