As I understand the term;

Bitt:
A sturdy post mounted on the bow or stern to which anchor or mooring lines may be attached.

Fo'csle:
The fore-deck, the forward part of a vessel. Also, importantly, that area of the ship holding the crew accomodation in the sailing ships. (Changed with the introduction of motor power).

Off-watch crew might gather on the fo'csle, above their quarters, around and perhaps seated upon the fore-bitts. Hence a forebitter, a song sung in the "leisure time" available to sailors. Not a work song (shanty).

AndyG