It's osage, mulberry doesn't have bark like that. Ring thickness has more to do with a tree having full sun and a big crown allowing it to grow fast.
And water. Mulberry does ooze milky sap, though. The wood does look very much like mulberry, but the bark doesn't at all. Dry it, weigh a chunk compared to the osage, and treat it accordingly.
EDIT: Whoops. Brain fart! It very well might be WHITE mulberry, rather than red. I just trimmed a tree, and the bark on the branches was smoother, but the trunk and secondaries had craggy bark like that. If so, white mulberry is great bow wood, too. I have made branch bows from the long suckers that pop out when people "top" their trees. They make for tough, stringy lumpy staves that are hard to break, and you can either make a sapling-sized bow with a crown from a 4" dia sapling/branch, or chase a ring out and treat it like low end osage.