He already owns this e-cigarette and it trying to figure out what kind it is to get some carts.
If that is really a 2-piece - when you buy a new pre-filled "cartridge" if it has screw threads (male of female) and screws directly onto the battery and that's the whole ecig then it is a cartomizer aka carto (not a cartridge aka cart). Cartridges are used on 3 piece models that have battery, atomizer, cartridge. Cartomizers are a combined atomizer and cartridge.
I don't think anyone has enough information to identify it. Even with the length and diameter it could be tough. Can you call the place where you got it and ask what company makes it or any other identifying information?
But yeah - if it is meant to be a disposable it would probably be hard to make it economical. The carto cap might be sealed with glue to make it hard or impossible to be cleaned or refilled. Read all of the sticky threads in the top section the New Members forum. The Library articles are especially helpful including the video links in it and the videos for broke newbies. Maybe it's time to pick a new model that has easy to find and inexpensive parts and carts or cartos available on the web, something you could refill with ejuice yourself to get more use out of each cart or carto. A second battery so one could charge while you're using one, etc.