Is this one circuit or several circuits. This can be caused by a down stream neutral to ground connection. That tends to happen in those over stuffed ceiling boxes as often as anything. You can't get all the neutrals under a big wirenut without some copper showing and it gets grounded.
As soon as you put a load on it, some of the current is diverted to ground and trips the GFCI. A regular breaker doesn't see anything wrong. They also don't like multiwire circuits but that usually needs 2 loads, one on each ungrounded conductor to make it trip.