The HW current fault means that the drive was being asked to provide more output current that it is designed to handle for longer than it can handle it. So for example, the PF400 is strictly a "Variable Torque" drive (in A-B parlance referred to as "Normal Duty"). That means it is designed to deliver 110% current for 60 seconds, or 150% current for 3 seconds in an I2t curve, or 180% instantaneously (100msec). If you attempt to exceed that, the drive will EITHER respond by reducing the commanded speed to reduce the load on itself, or if that is turned off in programming, it will fault. So we can assume it is turned off. Check the programming of A148, the OL Response function, if set to 0, that would allow it to trip if the current rises too fast. Also check A170, the Boost Select function. The default setting is a value of 4, which is for a VT load profile, but that includes what is called IR Compensation. That automatically allows an increase in the voltage boost at low speeds if the motor is cooler. Try setting that to a value of 5 which turns off the IR Comp, see if that makes a difference. Don't know why it would be related to the generator test, but I do know that IR Comp can sometimes make a drive trip itself under the right (wrong) circumstances.
That brings up this curious thing about your observation of it only being related to having been run from the generator 6 hours earlier. It's possible that this is coincidental, but it sounds as though you have observed it as a consistent event sequence. That raises more issues than it solves. I can't see it having anything to do with the VFD itself, although honestly, I kind of like Keith's idea on humidity. For example when the generator is exercised, the exhaust must be vented, so a lot more outside air is introduced to the room to replace it. Six hours later the pump station is being called for less duty and the VFDs shut down, cool off and the added room humidity condenses in and around the drives, current flows to ground and the drive is tripping on that current flow, or at least the sensors used to determine current flow are being shorted and giving a false signal. Remote, yes, but I think plausible.
So what else, if anything, is different about how it is run when in the testing of the generator takes place?
"You measure the size of the accomplishment by the obstacles you had to overcome to reach your goals" -- Booker T. Washington