What I would do before I would spring for a dozer and find out the swamp is to swampy, hire out a dozer or try your backhoe. If he makes it awhile without any problem you might have him do the whole thing and be done with it.
About 20 years ago I had to put a driveway through some swampy land and I had a guy with a D8 cat, I think it was, to do the work. I was really concerned about him getting stuck as he plowed into it. But he didn't hesitate and he didn't have any problems with it. I had 7 or 8 dump trucks hauling fill that had a lot of 6" to a foot in diameter rocks in it. The trucks would back down the driveway and the cat would just spread it around. After the boney type fill was brought in about 4-6" of road gravel went over the top. I never had a problem with it to this day and I would have bet I would have been sinking all the time.
I don't think it would pay to put dirt as fill, but stoney fill would work. Depending where you are, that might be expensive hauling. I had to haul about 50 quad-axle loads in. Maybe you don't have to go as thick as I did.
Then again, when somebody mentions swampy land, how swampy is that. It might be like quicksand or it may just need a foot or to of solid fill over the top.
If it were me and I had a backhoe, I would get a couple trucks of boney fill and start at the beginning and see how it goes. You should be able to tell how muddy it is before you get in to much trouble. Keep a winch handy and keep us posted.