Right.......except, when for instance did the Queen's Own Rifles get their name? When there was a Queen on the throne--Victoria. Had the regiment been formed when say King George the whichever was on the throne, they would have been called the King's Own Rifles, and that name would stick. So, back to the original question--what regiment would have been extant in 1795-1815 which would have had a QLI with the Q relating to a Queen? I do not find a reference to such a regiment. The Edinburgh or Queen's Regiment of Light Infantry was as you say a militia battalion, formed as part of the Royal Scots which is a very old Britishregiment from the early 1600's. That 3rd militia battalion though as best I can tell was formed around 1887, under Victoria's reign and they would of course have been using Martini's by then.
It may have been a local colonial unit I suppose, although those marks were usually branded on the stock, not the tang, or some local or old marking for a regiment which didn't exist for long. Nice musket regardless; it would be interesting if these old girls could talk!
Ed