If it is decomposing to the point of an odor being emitted, nothing will ever come from it.
I think that you might be better served by just letting it continue the way it has been going. The odor will subside because basically there was no odor until you disturbed the stump.
You could break it apart with tools a dispose of the pieces but that's going to be messy. Could you cover the stump with soil? That might hasten it's breakdown.
