Yes, it would be better to run a grounding probe. The GFCI will trip on a ground fault which in the absence of a grounding probe in the tank means it pretty much means it can only shut off if water gets onto a plug (ie., thus connecting one of the hot or neutral wires to ground -> thus "ground fault" -> thus tripping GFCI).
Without a ground probe in the tank, a submerged electrical device may have a leakage but without a path to ground the electric potential has nowhere to go. Thus if you stick your hand in the tank and say, your foot or something connects to ground (not the "ground" as in "floor" but I mean "ground" as in "somewhere for the current to go to") then you become the ground, thus getting zapped and bypassing the GFCI. Whereas a ground probe would be a better ground than you, and thus the electrical current would go there instead.
At least that's how I understand it.
