"Chanteys? One doubts it."

SHANTY. A small hut on or near a beach. (No "chantey" t'all.)

"... in the hope that it will introduce some better songs into the Service."

A senior Blue Light officer like Tucker would be all over the forebitters too. We do know sailors borrowed heavily from popular music: Heart of Oak; running with, and getting out of the way of, bulgines &c. That said, Tucker's glossary suggests instrumentals for the fife. Fiddlers had options.

Seems our boy went after the merchant fleet and general public as well:

"Everybody's song-book: a collection of nearly two hundred popular and national songs, duets, trios, rounds, and harmonized airs, set to English words."
[Same authors and publishers as before.]