Stumps are always a pain in the ass. There just isn't any easy way to remove them, particularly if they are from old huge
trees. If you had a couple of years you could drill the stumps full of holes and inoculate them with
mushroom spawn. You might even get a crop out of that if you found the right mushroom cultivar for the tree species and for your climate.
What is the soil profile like? if there is a layer of stone beneath the soil less than a couple of feet down, that might be your salvation. You'll need to excavate down till you reach that stone layer and then build your forms and pour your piers right on top of that. Once done, you'll back-fill around those piers --- or carve out your root cellar at that time.
But if you are going to that much work to excavate down to pour such
concrete piers -- why not just dig out the stumps? You'd need an excavator either way. Unless the stone is less than 2 feet down, you don't want to be hand digging your foundation.
Depending on the size of the cabin, I've seen people build with jacks attached to the corners of the building so that you can crank it up with a couple of turns every couple of years. These aren't cheep, nor are they very stable if you live in earthquake country (like I do). Basically, the cabin floats above the ground on these jacks. Every so often you go out there with a level and crank up one corner or another to bring the thing back to plumb. Its basically the same system as you'd have in a camper or motorhome, just a bit more permanent.
If it were me (and I planned on keeping that cabin for 30 years), I'd bite the bullet and dig those suckers out. $2000 spent now will save you a lot of frustration down the road.
Best of luck.