I cannot say that I personally used one or saw one being used. I do remember seeing one in the ship's armory on the USS Coral Sea in the early 1950s. I was drawn to it because I had an interest in the Trapdoors which was my first grown-up rifle in the 1940s. My older brother, also US Navy, told me that he saw one on the USS Gladiator, a minesweeper, in the late1940s.
It was a very simple cut-off barrel (12" +/-)with the stock shortened. No other changes that I remember. After having fired the H&R, I had no desire to test my shoulder with the steel butt plate of the trapdoor, although I was tempted.
When I was stationed at the Mothball Fleet in San Diego, one of my jobs was taking small arms off the old ships and cutting them in two with a welding torch. I kept hoping to find one of the Trapdoors to liberate, but I never did.
20 or 30 years later, I saw an ad in the old Shotgun News by a guy who was selling one of the Trapdoor line-throwers. I immediately contacted him and bought it. It was part of a set that included a barrel bracket to hold a cannister, a cannister, 2 spools of line, about a dozen projectiles, and a re-winding machine. After I got it, I made the mistake of contacting the old BATF to make sure I was OK in having a short barrel rifle. They told me I had to provide documentation that it was a legitimate US Government alteration otherwise I'd have to remove the barrel. I was a friend of Frank Mallory of Springfield Research Service and I asked Frank if he knew of any provenance. He found a couple of references indicating that XXX number of Trapdoors had been converted during WW 2 for issue to both ships and shore installations. I sent this information to BATF but they declined to approve the line-thrower because the information did not contain specific dates and serial numbers. By then, I was completely soured on the entire episode so I told them I would remove the barrel. I did not do that but, instead, sold the entire outfit to a collector in New Mexico.
Frank Mallory is long gone but you might be able to find information in his later published SRS publications, assuming you can find copies.
And that's all that I know about that. Old-man memory notwithstanding.
And Dave, I wasn't on the USS Constitution. I was too young at the time.