Why does a GFCI breaker trip when another breaker is accessed?
From the description, it sounds like the neutral may be shared between the two circuits, which you can't do with a GFCI. The GFCI is looking for an imbalance of current on the hot and neutral lines, so if the neutral is shared with another circuit, there will be some current returning on the neutral that did not go out on the hot.

If you can trace the circuit and find the connection between the neutrals on the two circuits, and isolate them, that would be the easiest and cheapest fix. If the neutral is shared via a multiwire branch circuit from the breaker and can't be split, you can change the gfci breaker to a standard breaker and install a gfci outlet after the neutral splits for the two branches. And I'm also seeing two pole gfci breakers which would allow you to replace the two breakers with one that provides gfci coverage to both circuits and should handle the shared neutral, which may be a good fall back option if you can't locate the shared neutral.