Agree. The ground rod shouldn't have anything to do with the GFI.
GFI is simple. Current is measured on the outgoing black wire. It is again measured in the return neutral wire. If they are different by more than 5ma, it will cut power.
Ground has nothing to do with it.
Think of a garden hose looped out and back. And gpm relates to amps. If you put 10 gpm in one end of the hose, 10 gpm will come out the other. If not, there is a leak. If there are tees in this hose, with other sources for water to be introduced, or to leave the closed system, then gpm in won't equal gpm out.
You have neutral issues. Circuits are crossed. A neutral from something on breaker 3 might be tied to breaker 2. Given the flow another path to take. (Think tee in the hose analogy). The GFI is sensing it and tripping.