I've tried doing this before and got tired of ruining good chains because of dirt and rocks that I couldn't see. One of the reasons that you're having a hard time cutting is that you are essentually trying to run the nose of your bar through one huge twisted knarly knot. It's by far the hardest part of any tree, and you always end up cuttings on the end of the wood grain no matter what you do (like you're trying to mill a board). Over all, the entire event sucks.
From my limited experince, I would say that you have 2 reasonable optoions.
First option is to burn it out. You can use your saw to cut come of the wood out if you want, or make some grooves that can be used to retain used motor oil, kerocene, or diesel fuel, all of which will eventually soak into the wood and MAY help it burn faster. I generaally flush cut the stump, and leave it for a year or two so that it will dry out some, then burn it. Get a good fire going on top of it. Us a bag of charcoal if you want. If you can, take a fan out and let it blow on the stump as it burns. It will sound like a jet engine, and it speeds the process up a fair amount. If you're going to cook over the fire (hotdogs, beans, or what ever) then I would skip the motor oil and such. After the first burn, which could last a few days or even a week depending on the stump and how long you countinue to feed the fire, you may need to dig out a little bit around the base of the tree to expose more of the roots, and then do another burn.
Second option is to rent a stump grinder. It does a MUCH better job, but I'm to cheap to spend the money. Dad rented one last year to remove the stump from a hard maple that 48" at some parts of the base. Took him about 8 hours total, and I think he hauled away 3 truck loads of saw dust. Now he has a flower bed there while the roots rot down over the next few years.
Either way you go, remember that in a few years, the roots that are left behind will rot down. If it's in your yard, that means you're going to have a hole form in your yard, and after hitting that hole with one of the wheels on your lawn mower (tractor) for a few weeks in a row, you'll start talking about it like it was your mother in law....
Just my 2 bits,
Mark