@JerryMc If you're going to use letter bits, you'll need to get a .375" reamer to go with the U size bit. You could just get a 4 flute 3/8" end mill (carbide) and use that for final size.
As for your crappy lathe 3 jaw chuck, make sure it can grab something 4" in size. IME, most 'crappy' lathes don't have chucks that can (I had a 7x14 that I don't think it could have grabbed even with it's "4 inch" chuck). "Plan B" could be placing the cap on something sacrificial, and simply clamping everything down to the German standard. You know, Gudentite.
I used my drill press for the first time in over a year this weekend. Drilling holes that needed to be tapped either 1/4-28 or 1/4-20. The collets for my mill (I'm using ER32 collets with an R8 holder, so I don't need to wrench on the drawbar every bit change) to make things easy. I need one more collet that didn't come with the latest 32nd increment set I bought (need the 7/32, not in that set, one is on it's way now). That should hole either a #3 or #7 drill bit.
You'll want to have your drill press spinning as slow as possible. <200rpm is good. <100rpm is even better. Use either cobalt or carbide drill bits. If you spin too fast, you'll case/work harden the 304SS cap and it will be a pain to try and make a hole in it (if you even can). Anytime I drill into (or machine) 304SS I use the 90rpm speed on my mill and work it slow. 4 flute, carbide, end mills are typically what I use. I've dropped not a small amount of $$ on cutting tools. Not to mention work holding, tool holding, and the DROs on the machine. Probably going to get another item for the mill that could be a bit up there in cost too (depends on which option I go with) but it will make setting up on the mill easier/faster.