You could use the 1" pipe and grind down the cap to 1 3/8" or find a thinner wall cap if they exist.
A 1-inch coupling would probably work pretty well without grinding it down. They are usually just cylindrical without the boss on the end that a cap would have. If you're buying a whole stick of pipe (21' long) from a wholesaler, it would come with a coupling on one end. At least they did 40 years ago. But a coupling isn't that expensive if you need to buy one separately.
