I do mine in a CNC using coolant and only the finish tool. I'd recommend coolant if you can. The lathe is 15" between centers (thereabouts) and it works wonderfully. I begin by doing 1/2 of the barrel, then I make a 1.5-2" alum tube that fits the barrel taper I just cut. This is OD turned, then ID bored to fit. It snaps on like a morse taper, then I turn the barrel around, chuck on the alum, and do the other half. With your taper attachment, you can do the same thing. The fit of that tube does have to be reasonably close. Make sure you put good centers in the barrel before beginning. Obviously, I use a live center in the unsupported (not in the chuck) end. I also use that tube trick for any crown work.
The inserts I use are a 135 Deg standard finishing insert, with a .015 radius iirc. With careful indicating on the part, you should be able to have a transition that cleans up with a file, and after sanding the barrel, is invisible, even when looking right down outside of the barrel in the sunlight. With the barrel chucked, and only 1/2 of it being worked at one time, chatter is a non issue. Well, it has been so for me anyway.
And no, you won't hurt the barrel one bit. The barrel maker turned it to begin with. If it ain't crooked now, it won't be when you're done, unless you get it hot as blazes when doing crappy turning work. Mine make real nice, tiny gray chips when I do them. I would guess I did about .050 or so per side depth of cut. Feed was whatever the lathe chose, and rpms were not very high. Probably in the neighborhood of 200 or so. The feed was enough to break the chips nicely, and get the heat out in them.