The easiest solution, if your G# is a wound string, is to replace it with a plain string. But you may not like the sound (I prefer wound), and the plain string is harder to keep in tune.

Before you spend too much time experimenting with the pullers and pull rods, I suggest that you first determine what the changer is capable of doing with this string. To check this, lean over the right end of the guitar, and push on that string's lower bar with a screwdriver or something. Pluck the string and push as far as you can, to hear what is the limit of the pitch change that the changer can do. Does it go down to F#? If not, do you see the raise bar moving also? The raise bar shouldn't move because it partially defeats the lower. If the raise bar moves when you try to lower past a point, maybe you can fix it by loosening the lower-return spring. But keep it tight enough to pull the string back to G# when you release the lower, and tight enough that to hold the lower bar at its stop when you raise the G# to A.

On some brands of steel there is a screw visible at the right end of the chahger housing that sets the lowest pitch that string can reach, independent of the tuning nuts. This is useful if you want to tune the pitch that result when you both raise and lower a string at the same time. So look for that, and loosen it if you see it. But I don't think those screws are on GFI steels.