Ned, the protagonist of The Private Misadventures of Nell Nobody, uses a Ferguson rifle.
This is where the novel veers off from reality, because the Ferguson was prototyped and first used during the American War for Independence. It was a breech-loading gun (it still only fired single shots, but it was faster to reload through the breech as opposed to the muzzle). It was also shorter, and lighter. But as with any prototype, there were problems with it, probably the first and most important being that the breech-loading mechanism would jam. You could still fire the weapon, but at that point you had to load it at the muzzle like a musket, which defeats the purpose. It was also expensive to build, costing four times as much as a Brown Bess, and there were only four gunsmiths who could make it.
By the time of the French Revolutionary Wars and the Napoleonic Wars (the ‘Coalition Wars’), the British Army was looking for a better alternative, which they found in the Baker rifle. That is more likely to be the gun they would have used, if they had even been using riflemen at all during the period in which the book is set; the Experimental Rifle Corps was not formed until 1800.
I have never fired a reproduction Ferguson rifle. I’d like to, just so that I could experience what my protagonist does in battle (erm, without the battle part, preferably). The other marines would have used a sea-service musket, which was essentially a Brown Bess that may or may not have had the barrel cut down a few inches more, and was made of an alloy that wouldn’t rust overnight the way an infantry musket will.