Let me put this all together.
You'll need that barrel to have a working clarinet. Full stop. No barrel means trying to find one on ebay or Craigslist.
The horn you have doesn't appear to have a sterling silver bell or even silver plate. That doesn't necessarily mean "bad clarinet," but it probably means "not a valuable clarinet." If you want to
post more pictures, someone here may know who actually made the clarinet.
If you're wondering if "Delmonte" made your clarinet, the answer is probably "no." It's probably just a store that sold the case or clarinet and case. Back in the early 20th century, this was extremely common.
If you want to get a mouthpiece that is made by the same company that made your horn and around the same time, you're probably making your life needlessly hard and you'll spend too much money. The only thing that you really need to find out is if you have a C, Bb, or A clarinet, then get a mouthpiece that you like. Or you can try to find out the manufacturer of your clarinet (I'll make a wild guess and say Conn), find out when it was made (probably 1930s/1940s), hit up some mouthpiece forums to find out what mouthpiece came with your horn, then troll ebay, Craigslist, etc. every day to see if you can find that mouthpiece. Then you might find that the mouthpiece is too difficult to play on and/or is more expensive than your clarinet is worth. No sarcasm.
Additionally, your clarinet looks like it needs, at least, all new pads. Add a couple hundred $ to how much you're thinking of paying.
Gandalfe owns some really pretty Selmer silver plated (I think; I don't remember him saying sterling) clarinets. They are most definitely professional instruments and are worth a lot. I owned a Conn Pan-American metal clarinet that looked a lot like yours. Definitely not a pro horn, but it had a nice sound. I paid somewhere between $5 and $20 for it in the 1980s. You might search ebay and find a horn that's identical to yours for around the same price, even without adjusting for inflation. I wouldn't recommend one of these to a beginner, but if a student came into my office with one in good shape, I wouldn't turn them away.