Well, John Peel was a fox-hunter, so no guns involved there; if poaching songs are excluded, then I can't think of a single traditional song about shooting per se (The Fowler is rather in a class of its own).  Plenty with brief references to going out "with my dog and gun" but only as an oral-formulaic commonplace, usually.  I don't doubt that a good few were written, but I suspect that they didn't make it into tradition because ordinary people didn't shoot for fun, but for food.

Having said that, a search for "shooting" at  Bodleian Library Broadsides  finds a few examples, though probably not many that you'd find useful!  Here are some:

The Morpeth Sportsmen  In which an incompetent shoots, first one of his own dogs, and then two pet ducks, for which he is beaten up and humiliated by a local farmer.  Printer and date unknown.

Going A-Shooting  Printed between 1813 and 1838 by J. Catnach, 2 & 3, Monmouth Court, 7 Dials, London.  Another catalogue of disasters.

Sportsman, or, Dog and Gun  Printer and date unknown.  This one seems to fit the bill, though no tune is specified.

The Sportsman, or, Dido and I  Printed between 1780 and 1812 by J. Evans, Long Lane, London.  This one seems to fit the bill, too, though again no tune is specified.

The Bonny Moor Hen  Printed between 1863 and 1885 by H.P. Such, 177, Union Street, Borough, London.  Not the Jacobite song, but one in which a group of miners fight a pitched battle with the gentry, who are trying to stop them shooting game.

Out with my gun in the morning  Printed between 1850 and 1899 by T. Pearson, 4 and 6, Chadderton Street, Oldham Road, Manchester.  Not really about hunting, but rather a general celebration of "simple" country life.