then it seems like the fault must be on the neutral side of the circuit. If it was in the load, a different load would not trip it. If it was in the wiring on the hot side, it would trip with no load. The neutral side is in theory at the same potential as 3rd wire ground, but carries current when there is a load, which could leak through the fault to ground.

One tool I have found very valuable when diagnosing fuse-blowing faults is a sensitive clamp on ammeter, such as the Extech or Anchor products. Both have a min/max function which will capture the maximum current seen until reset. I don't know what the sample rate is but it seems to be high enough for this purpose: you clamp in on suspect circuits, enable the circuit (which blows the breaker) then see if you got a momentary current spike (indicated by the max reading of some amount greater than zero) in the suspect current. If everything is off, you should read zero unless there is a fault. Repeat until you find the fault. You can use this on your ground wire an follow the current from the panel towards the load. These meters will only read down to about 10 ma, so if you have a very high resistance leak it might not indicate but still blow the breaker. Usually though, a fault is much bigger than that. Unlike voltage, the current will follow the circuit back to green ground, you will not see it in branch circuits that are connected, but not carrying the current.