If you are serious then the only chance you stand and it is an outside chance at that is to drill a large number of 1" holes into the top as deep as you can and pack them with potassium nitrate fertiliser. Leave for about a year refilling as necessary and then after the longest hot dry spell you get build a fire on top and cross your fingers that the impregnation of the nitrate is enough to make it smoulder away until underground. It may still be too wet in which case you are stumped.
This sort of works but it also primes the wood for fungi to take hold and I got a spectacular chicken of the woods on my former pear tree stump. The ground where it was is still subsiding as the deeper roots were not eliminated by this method. In the end I prized the last bits up with a scaffold pole using an axe to cut the tap root. YMMV
I'd tend to favour digging it out.