I'll have to check the copyright on the photo, Mark. It's in the latest edition of Stenson's bio, though, so not too hard to find.
Lightoller knew at least one of the Carpathia's deck officers beforehand (Dean), so it's possible it was handed over from one mate to another, or as a gesture of solidarity. I don't think the image is clear enough to make out the cap badge, which would give clues. There certainly seems to have been a sense of the sea-faring fraternity about the whole thing - Bissett made kind comments about the surviving officers, and they were presented with signed photographs of the Carpathia's officers.
Some of the uniform items are still extant, but most are missing - Lowe presented at least one button to an American he got along with particularly well, and the family gave away others. The hat is now, unfortunately, lost (Harold W G Lowe did look for it as an adult but couldn't locate it). Other items unconnected with the Titanic disaster still exist - Lowe's RNR dress uniform bicorn, Pitman's RNR uniform, James Moody's uniform buttons from the Conway etc. Of these, only the Pitman items are currently on public display.