It's unclear to me where the boat is located. Guessing that it sounds like it's in its slip. That makes things a bit more complicated. I'd first do a bottom survey to know how deeply the keel goes into the mud, what mud consistency, and where will the boat be moving to in order to know if there is enough water to move it without it getting stuck again elsewhere.
Without that survey, it's hard to know what method to use. You need to know how much flotation you need and where you are going to move it to. If you use a crain to pick it out of the mud, where are you going to put, first the crain, and second the boat?
If the boat is in the slip, you likely can't lean the boat over and move the boat at the same time - too many things in the way.
Lightening the boat is good, but you need to know how much you need to move the boat upwards to get unstuck. Adding flotation is probably the easiest given the constraints of being in the slip. However, mentioned above using ratchet straps is not how that is typically done. You sink the plastic barrels full of water, attach to the hull with the bottoms facing upwards, then you use compressed air hoses and fill the barrels with air pushing the water out that way. That way you can make a solid connection to the boat with the barrels in their position and you can control the flotation better by controlling your compressed air flow. They make flotation bladders but they are pricey. Cheap plastic barrels are easier.
But no matter what is done, you need to know both how much flotation is required and where you are going to move the boat. It's a real drag floating the boat in the slip only to get it stuck somewhere along the route...
dj