The important thing is to get it out before the neighbors see it.
Growing up on a lake as a kid, I remember when they would drain/dredge it empty.
Backhoes would make mats of railroad ties and create their own pathway across the muck, raising the mat behind the backhoe, bridging it around to the front, and dropping it in front, making it's own road as it moved along.
Id go back into town and have somebody get a ****load of 4x4's (whatever is around) and run them under the FEL bucket to give something for the bucket to lift against and un muckify the front end.
I'd also run 4x4's etc crosswise behind the rear tires to give it something to ride over as it goes back.
Same for the front if you can get the bucket to lift off something solid.
The rear box hopefully will lift enough to get it clear, but I'd be trying to get at least plywood under it so it doesn't get it's arse stuck in the mud too during all the other excitement.
I can almost see the tow cable snapping and slingshotting it's arse through the truck rear window like all the youtube videos. Unless it's an easy pull, I'd be thinking, up, and then across the mud, rather than through it.