Don't restrain the wild GCS that's my advice from having one once. Have a lever and floodgate with a cage trap on the other side so you can recage the GCS when required. I had FBs coming up then too, and I believe the GCS never targetted them. Dropping a kitten onto the cage area would be better to prime the traps for FBs. MY design was basically like this, but a bit bigger :-
######D##
##FPHPccc
cXC~~~ccc
##FPHPccc
#########
# = Wall
~ = channel
F = Floor
D = Door or Doorway
c = cage trap
X = flood gate
C = GCS cage
P - Pressure Plate set to creatures
H - hatch over a pit
I put a pit / hatch above the main cage trap area, to drop tame animals in. FBs should enter from the east, and try and path through the north doorway, past the traps. the pressure plates are all linked to hatch next to them, fooling the GCS that it can path to the cages, so that it shoots.
The floodgate is opened to lure the GCS back into a cage. If you put a GCS on a chain by some miracle, how would you recage it later? Note, since the traps are to be-prewebbed, you don't need the GCS here all the time, you an run a silk farm as well on a similar basic design, just ensure civilians can't try and collect it from the trap area, while still allowing pathing for invaders - maybe using a variant of the Pressure Plate / Hatch combo, but set to trigger for both dwarves and creatures, should the get past the cage traps.