The information at The Fiddler's Companion seems to be taken from Williamson's book, which includes a good bit of speculation mixed with some history:

Burning of the Piper's Hut.

Neither pipes nor pipers were banned in the aftermath of the Jacobite rebellions (or at any other time, though certain clergy of both Catholic and Protestant persuasions did a good bit of freelance instrument-burning on religious grounds); since that seems to be the sole basis of Williamson's suggestion that the tune might be of the period, it should probably be disregarded for now unless anybody has any actual evidence of a date. More likely to be just one of those tune names that commemorate a local incident now forgotten, I'd have thought. Nigel Gatherer quotes Williamson and gives the tune at  The Burning of the Piper's Hut