I am a retired forester and have considerable experience with this problem. I suggest that you first check with your local extension forester. Here are some things to consider:
Soil: If there is much clay and if the soil is not dry, tractors can compact the soil. Compacted soil does not allow air to reach the roots and water does not infiltrate. Roots need air. Water that doesn't go into the soil goes across it, and that's how you get erosion.
Dozers: In our soil in western Oregon, 6 passes over a spot by a dozer, if the soil is not almost bone dry, will compact the soil. A dozer piling logging slash works back & forth, back & forth and does a great job of compacting the soil. The vibration of the machine helps, too. Dozer piled slash usually results in 30-40% of the surface area compacted.
Tractors: The ground pressure of a tractor is higher than a dozer & will compact it just as bad.
What works: Hire a track mounted excavator (track-hoe, back hoe, all the same). Working with a bucket & thumb, they can get the job done faster and with little or no compaction--less than 5% in our area even when the ground was wet. The bigger the machine, the better, as long as they can fit between or go over the stumps, as they can get more work done from one spot and don't waste time moving when they should be piling. They can use the teeth on the bucket to rake the ground to remove the weeds. Excavators build better piles than dozers or tractors, because they put less dirt in the pile, can build the pile higher and can use the bucket to compact the pile so it burns better. They can also be selective about what they pile, in case you want to leave certain stuff--with a dozer, everything in front of the blade goes into the pile. They work faster than dozers and therefore are cheaper.
Burning in place (called a broadcast burn in the industry) also works, but you need a crew and fire equipment to prevent or control an escape fire. There are businesses that do this, but maybe not in your area. You need a wide line around the burn area, say 4 ft. wide, as a fuel break. A dozer does a good job at this and the compaction created is a small part of the total area.
Burning is fun! Whether burning piles or broadcast burning, it is a lot of fun, at least for a pyro like me.
Whatever you do, be sure to control weeds afterward, or you will be right back where you were before, which is a mistake I made & knew better but got distracted at a critical time. So stay on top of things with your sprayer.