Root plow with a dozer... then rake it. Disking will help a lot but will leave some.
If you do it with your own equipment, you can foam fill the tires... atleast the front. It can happen to the rears no doubt, but in my experience the front tires get the most flats.
No matter how you work the ground when every thing starts growing again get on a good spraying regiment. You can spray a lot of country for the cost of tires, diesel, foam filling, $15K shredders, and the time it takes to do mess with all that. Not to mention the fact you just manipulating the brush... not getting rid of it.
You could always just leave the stobs and start spraying when they start growing. That keeps you and your equipment out of it totally.
IMO 4" is a good thing. Most of those will fold over when you drive over them. Its the 2" ones with the angle cut from a bush hog that get you because they have no give.