We had a native tree in our backyard in Virginia, and I recall the berries being a bit larger in diameter than an eraser, and 3/4 to 1" long. Like a stretched blackberry. It was a mature tree, though. I don't know if fruit size is normally small in immature mulberry trees.
Have you considered grafting a few different varieties onto it while the tree is still young? Whether the native turns out to be a winner or not, there are many prized cultivars that inspire mulberry freaks who are almost as odd as a few fig lovers I know.
Good luck