Could you line the bottom of the oven with foil?

Leave the stone in the oven and run your clean cycle...it will smoke like crazy, but it will be clean.
Never wash a bake stone with soap....because ever after evry thing baked on it will taste of that soap.

You can put a cookie sheet under it or even just a sheet of foil to catch the grease. It really sounds as though it was treated with too much grease or is it just really greasy food?