Nothing is going to do it fast...
When is the burn ban over?
You can also, do this every night when the weather is good, nightly "camp" fires.
For now, I would dig the deepest / biggest hole in the ground that I could with your tractor (but you can use a bulldozer, too

). Save the dirt to spread out later. Start filling the hole with the dirt coated mulch that will not burn... drive on it and pack it down if you can.
If you fill the hole, continue to pile it as high as you can. Dirt with the wood will help it rot faster. Once you are done filling and piling up the brush, cover the pile with dirt, plant some plants on it and make it a "feature" and let it rot down over time.
If you have a view you want to block (highway or neighbor's house) put it there. I've piled stuff like this over an embankment to create a flat spot. I've seen it done in ravines but you loss the ability to "collect" the top soil to be used elsewhere.
Get a heavy duty landscape rake on the back and the grapple on the front. One thing I don't like about the ratchet rake is that it ties up the FEL. Even a box blade with rippers on the back would help gather up the mulch.
Good luck with your continuing project...