|Johan Anton "August" Menke on his 90th birthday. He came to America when he was six years old.|
I finally found it! Ancestry has a collection titled "Bremen, German Deaths of Sailors and Ship Passengers, 1834-1875" and in it I found this document:
Note the date is September 27, 1871 when previously we had noted their voyage took place in 1872 but have not yet found the ship's passenger list. We can be sure it is him due to his wife's name being listed correctly, Elizabeth Menke geb (born) Geers. It also notes his birth location as Schwagstorf amt Furstenau and his residence as Bersenbruck, which is what we have noted for him. It lists his cause of death as tuberculosis.
The death record appears to be an index of people who died at sea and it does not list the ship's name so I'm confident this is a secondary source. Still, it is the best documentation I've found so far for his death. At least now I have a date range to search for the ship's passenger list. Little by little, progress is being made to document their voyage to America.