The brass on 375 is a bit thicker than the 38-55. The 38-55 pressure is about 30,000 psi and the 375 can be 52,000 so it would probably be best not to shoot the 375 in an older gun. Old black powder barrel are usually .379 groove but I have seen some that were .381. Modern ammo usually has a .377 diameter bullet to work in both new .375 and old .379 barrels. Barrel liners, there is one in .379 with a 1 in 18 twist and two with .375 groove, 14 twist and 18 twist.
A Douglas barrel blank can run between $250 and over $300 depending on length, plus shipping. They can do a round contour but not octagon. There are a few barrel makers that can do octagon.
Rebores, old black powder barrel are not very hard and most will cut rifle very well. In 1895 Winchester and Marlin came out with nickel steel barrels for smokeless powder. Some of these barrel are very hard and some not so hard. Never know how well they will turn out with a rebore but I bet that JES is probably good at getting good rifling. I have had a few that didn't look good so ended up installing a liner. If the barrel has rust pits it can be rough on reamers and cutters. I had one that sounded like the cutter was being pulled through gravel.