Helen,
If someone was an Officer in the Royal Navy they would normally appear in official publications with the letters RN after their name. If your ancestor doesn’t have that designation, he was probably a merchant navy officer. To which I would add that RN officers were usually permanently on the move whereas if your ancestors were mostly between Belfast and Cumberland then that also suggests a merchant navy background. You might find something on these sites:
https://www.crewlist.org.u
http://census.nationalarchives.ie/search/cl/home.jsp
Bear in mind that Belfast to Maryport was (and still is) a short domestic journey so the requirements to produce crew lists for the authorities etc were not the same as for international journeys. Records may be sparse.
The term Master Mariner also indicates a merchant navy background.
You are interested in the cause of death for James Brown in 1855. You could order a copy of his death certificate. The only likely death I can see for a James Brown in 1855 in Cumberland was registered in Cockermouth in Apr – Jun quarter, Volume 10b page 307. You can order a copy of that from the Passport Office in England. Order a pdf copy (ie a photocopy) as that will save you £4. (sterling). The current price for a pdf copy appears to be £7.
https://www.gro.gov.uk/gro/content/certificates/contact_us.asp
Here’s a link to James Swan’s death in 1906. He died from drowning (probably accidental) and there was an inquest:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1906/05573/4565013.pdf
PRONI may have the inquest papers but there is a detailed report of the inquest in the Belfast Newsletter of 2md January (attached). It seems a there was a strong wind and he may have been blown into the dock when re-boarding his vessel.
Have you searched Newspaper sites to see if any of these Officers were otherwise mentioned?
There was always a lot of commercial traffic between Belfast and Maryport and other ports in Cumberland like Workington & Barrow. Cumberland had a lot of coal, and so there was lots of employment in mining, steel making and associated industries. Cargoes such as passengers, steel, iron and coal went back and forth all the time.
