Deaths are registered in the district in which they occur - so you won't find a normal registration for servicemen/women on active service in WW2.
He is listed under the Armed Forces Records (index available on FindMyPast).
GRO WAR DEATHS NAVAL RATINGS INDICES (1939 to 1948)
vol 10 p3567
Arthur S Smith, Steward, HMT Lord Selbourne.
You can use those details to order a certificate from GRO (as an Overseas event) , but it is very unlikely to tell you anything you don't already know. They are usually very vague - the location will probably say "North Sea" and the cause will be "enemy action" or similar.
A better option may be to order a copy of his service records from the MoD.