Yeah, Mulberries are definately tough. I cut a few down on the edge of my property and all of them sprouted back from the base, growing new shoots at rapid speeds. I need to spray them.
From your description, I would 1st look into leaving selected parts of the property for bedding areas. Sounds like they're doing a great job of cover for that. Figure out the best areas of your property for it and the right size. Then, I'd probably come in with a dozer and clean up a descent amount of the rest and plant a recommended hardwood in place.
I hunt a 230 acre property that was heavily logged about 6 years ago. At 1st there was a lot of growth of invasive weeds, but now it's mainly junk tree's & cottonwoods growing up. Much of it is too thick to walk through, but created awesome bedding areas and cover for the deer. A couple years ago I went through a lot of it and opened it with a dozer, just enough where the deer can move through it a lot easier. There are still the thick area's but when the deer can move between them easier, I noticed more deer traffic. Not to mention, the place is a lot easier to hunt.
I'm sure the district forester will have some good advice to give... Good luck!