Be careful with mulberries, I had one in my last place (not sure if it was native), and it dropped 10lbs+ of incredibly soft berries, so effectively black dye into my back yard daily for several months a year
The white - look for Shahtoot - has long and slim fruit, with a much more delicate flavour. Can recommend.
Birds left ours alone for a decade, but once they discovered the fruit, they live in there from November through December (AU) relentlessly picking off the ripening ends of each fruit.
A bigger concern with mulberry trees may be their allergenic properties.
White has a cultivar named Morus alba 'Fruitless' that is often used in cities. You can ask for it if you want a tree that will not stain the pavement.
The normal tree with fruits is a birds magneto but is a medium-big tree. Use it if you can provide the space necessary.