We've got similar situations, except that I'm a bit farther along now than you are. I've got a small, two wheel drive tractor with a FEL. When I first got it I tried to move some of the clay dirt I had stacked for over a year. All I could get is about two shovels full. Through trial and error I learned. The first thing I did was to add a good=heavy box blade. That helped a bunch. Then I made a tooth bar to help break up the clay. A little more help, but that big stack of clay was still formidable. Then I started adding weight to the rear end. The more I added, the better it got. I even filled the rear tires with water. Now I've got around four hundred pounds of weight, plus the box blade, plus the water in the tires. I've got good traction with all that weight and it will safely carry a full bucket of dirt if I can fill it. It's still not easy, but with practice you become better and better with your technique, and the tractor works better and better.
A few tips I've learned over the long haul. When digging into the pile, if the bucket tends to buckle downward, you've got it tilted to far down. If you go to high up with your approach angle, the bucket will slide rise to the top and slide with little or no dirt in it. In really hard packed clay I try to angle the bucket for a fairly staight entry and set the height for scraping about and inch or so off the top. I may need to take for than one approach to fill the bucket. Once I do fill the bucket it's sometimes so heavy that I bury the front tires and the rear tires just spin. That's when I try the differential lock. If I'm still stuck nothing to do but dump the bucket and get out. That's when I turn around a go to work on the approach path filling holes and such. I've also found that the path works better the more level I keep it. The worst situation is when I dig a hole that the tractor has to climb out of with a full bucket. It's a better idea to have a slight downhill path with a full bucket if you can work that out.
I use low range in the transmission and high range on the differential. If I go to anything lower in the differential it's too slow. I need a littel speed for momentum to dig into the pile sometimes and the tractor has plenty of power to spin the tires in that speed selection. I normally set my engine speed to around 1000 RPM's and use the pedal to vary speed as I approach and dig, and then carry the dirt. It works well for me after some practice. Remember, nothings takes the place of practice. You and your machine will work better as you get to know it.
It's a big job and you might even consider renting a big machine if you need to move a bunch of it at one time. I'm moving mine a little at a time so far. When I need to really get it done I'm going big. No need putting all that wear and tear on my little tractor. It will last a life time if I use my head and take care of it.
Good luck with it and be safe.
Tom