The older Seneca falls were tooted to fit 3/4 so you might try a 3/4 something to see if t will just go. Still, that may be small for your needs. Good to get the How To Run A Lathe book and begin learning. Spending an extra few hundred or grand to get a well-equipped lathe may be worth the extra bucks because as you grow in your project likely you may wish to do more than your current thoughts.
A 1036 lathe or better, with gears or a quick change, a 3 and 4 jaw, steady, tailstock, taper attachment, tail drill chuck, fish gauge, a set of thread wires, micrometer, indicator with a holder. and a digital caliper. likely 3 grand plus for all this buying decent used.
You might measure a few guns to see what may fit through what spindle hole.
Yes, one can offset the tail to get some taper.
Do look at some YouTubes to see how barrels are turned and chambered, there are a few methods.
Agree an over 1" through-hole would be good, A SB 10 has 1 3/8 ( I think) and some a larger bore.
Some older smiths would put the muzzle in a 4 jaw and the out end in a steady, and perhaps run the OD between centers.