Yes you do need to file the GAR.
Terrorism Act 2000 Section 121
“the Islands” means the Channel Islands and the Isle of Man,
So "The Islands" includes the Isle of Man.
Terrorism Act 2000, Schedule 7, Section 12 (1)
12.—(1) This paragraph applies to a journey—
(a) to Great Britain from the Republic of Ireland, Northern Ireland or any
of the Islands,
(b) from Great Britain to any of those places,
(c) to Northern Ireland from Great Britain, the Republic of Ireland or any
of the Islands, or
(d) from Northern Ireland to any of those places.
Your flights fall into category C and D, so this paragraph applies to them.
Terrorism Act 2000, Schedule 7, Section 12(3)
(3) Where an aircraft is employed on a journey to which this paragraph
applies otherwise than to carry passengers for reward, the captain of the aircraft
shall not permit it to call at or leave a port in Great Britain or Northern
Ireland unless—
(a) the port is a designated port, or
(b) he gives at least 12 hours’ notice in writing to a constable for the police
area in which the port is situated (or, where the port is in Northern
Ireland, to a member of the Royal Ulster Constabulary).
(4) A designated port is a port which appears in the Table at the end of this
Schedule.
(5) The Secretary of State may by order—
(a) add an entry to the Table;
(b) remove an entry from the Table.
You have to give 12 hours notice to the police (which is done via a GAR) unless your airport is designated. Ronaldsway on the Isle of Man is designated. So you may have an out, if your airport in Northern Island is designated, but your handling agent at Ronaldsway will probably insist on you giving them a GAR anyway.
I hope that makes it clear.