I think we're talking about a couple of issues on these - (1) width at the breech and (2) barrel profile.
The problem with a straight 13/16 straight barrel it that it is harder to keep the wrist area proportioned correctly - it can be done but do it carefully. In some of the books they talk about tapering the lock bolster to kick the tail out more - the problem I have with this is that they don't mention that when you do, you are not just kicking out the tail - you are pulling the nose of the lock in as well, and this can lead to mainspring/reamrod channel clearance issues and not enough width at the front of the lock panels for good definition.
So if you want to give yourself enough width for good wrist shaping, a rifle with a relatively wide-breech area is easier to work and thus a swamped barrel will help with the lines.
I have seen a number of mountain rifles of the post-1820s era that have swamped barrels with pretty severe taper and flare - the problem is that the modern outer profiles are usually different, as Gary pointed out. On the originals, the skinniest point on the barrel is sometimes noticeably farther back than they place it now so it is less noticeable visually. In other cases, like Mike pointed out, the flare is sort of a "blob" right close to the muzzle - I have been told of examples sometimes even wider than the breech. Take a look at the barrel making techniques shown in Foxfire 5 and John Rice Irwin's book - these late period swamped barrels were being made by mountain gunsmiths and you would expect a lot of variations. But in general, the barrel wall thickness is usually much heavier relative to the bore size than most people like today - imagine a swamped barrel, 1 inch or more at the breech and not much less at the muzzle but in a 32 to 40-some caliber - much heavier than most folks find comfortable today.
So I guess if you go with the A weight swamped, it will help your wrist profiling. It won't really be representative of what you usually see on these types of originals, but then again really any barrel you use will not be correct if you go with something of the weight and handling characteristics he wants. And if the 38 inch, shorten up the area between the breech and entry thimble - this will help the lines so it does not look stubby.
On the other hand if you go with the 13/16ths straight barrel, just realize you need to proportion everything down a little bit to keep the right look - wrist height, triggers and guard, etc. I think there was one of Ian Pratt's guns on the Contemporary Makers site last year made with a skinny barrel like that and it looks great. I have seen them even made with 3/4 inch straight barrels and it can be done nicely.
Good luck
Guy