AIR 10/4038 (1942) and 4039 (1944) both refer to it as RAF Barrow.
Works - AP 3236 calls it Barrow. The RSP 4405/44 is 'Barrow'.
RAF Flying Training and Support Units refers to No.3 AGS as 'was to have formed at Barrow (Walney Island), but describes 10AGS as 'Barrow'. It also calls the post-war Gliding School , No.188, Barrow.
Perhaps there is a clue somewhere in one of those ORBs.