If you have a small BH bucket then put it on as you can exert more pressure on roots with the small bucket than a large one. Set up and dig as close to the tree as you can and still be able to cut/break the large feeder roots, if you have to move out 3 feet from the trunk to sever the roots that is ok, you can come back and use the hoe to loosen the soil under them. Reposition and dig out another side the same way working all the way around the tree, then start digging out from under the feeder roots working your way into the tap root (if it has one), once you get it dug out all the way around, use the hoe to push it back and forth wiggle it as much as you can then did some more till you can finally tip it over with the hoe and drag it out. A 15" stump shouldn't take over a couple hours to dig out, just keep at it.
Most important, call before you dig. Utilities might be ran thru the area, its a free service to call and have them mark everything but costly if you damage something without calling for a markup, you may have to pay the cost of repair.