I'd imagine that the best bet is they would have carried some type of smoothbore. Coming from Virginia, perhaps an English style fowler. These were the most practical of guns since you could use shot and large round ball out of one gun for both fowl or large game. A trade gun or military style musket is another option.
If you'd rather have a rifle (and I'm sure there were some), some kind of generic Lancaster style rifle might have been used. I asked questions about what an "early Virginia" rifle would have looked like, and the answer seems to be that no one really knows since so few survived and the ones that did are largely un-signed. Check out this thread for the details on that discussion.
Early Virginia rifle thread
If you've got access to a copy of Shumway's Rifles of Colonial America, there's a section of rifles of Southern Origin in Volume II, and some others that might fit there as well as anywhere else. I'm a big fan of #114 from Volume II, also seen here
Free Born Rifle
Also, there's the Johann Faber rifle, which I can't find a picture of right now, but which they believe may have been carried by someone in Daniel Morgan's company of riflemen and is an early southern style.
The bottom line is that at that early of a time for Tennessee, most of the guns came from somewhere else and most didn't survive. To me that means you can have some leeway and pick something you like, within reason!
BTW, I stopped in at Tip's shop this afternoon on the way into Nashville to see my family. He certainly seems to have sold a lot at Dixon's - there were only about 1/2 of his racks full! Good for him. He had some good-looking "Virginia-style" rifles there. I recall a really nice looking one in walnut, sliding wood patchbox, .58 C-weight barrel, 44" long. Can't recall the price, but it just needed finishing, I think.
Anyway, let us know what you settle on. Growing up in Nashville, and my mom's maiden name being the same as James Robertson's I've also got a connection to Nashborough!
