I don't know that you would have to change the cylinder. But that's the problem because I have no way of knowing for sure if the present cylinder is good to go or not. If it unquestionably O.K. then it shouldn't be changed, but by the same token, it isn't necessary to change the barrel, although it might be desireable.
Understand that changing barrels isn't a case of clamping the barrel in a vise, putting a hammer handle through the frame's cylinder window, and giving the frame a twist. This is an excellent way to warp the frame, and if you don't the new barrel is unlikely to just screw in and have the front sight come up at 12:00, and have a correct cylinder/barrel gap at the other end.
Also the barrel you have may have the front sight turned a hair to correct a windage error. Before making any judgment you should target the gun and see where it shoots relative to the point-of-aim.
I haven't checked lately, but they did have some .38-200 barrels in the 5" length. That said, while the 5" length was the standard one, during the war S&W made the .38-200 with barrel lengths of 4, 5 and 6 inches. With a war on they didn't hesitate to use anything they had that might work, or could be made to work. After the war, Numrich (The Gunparts Co.) bought literally tons of surplus parts from both the U.S. government and the S&W company.