GFCI tester test + GFCI = tripped GFCI
A CHGF is a GFCI and circuit breaker in the same package, using the same breaker contacts and trip mechanism. So, when your inspector pushed the test button on his receptacle tester, shunting a bit of current to ground, the GFCI-breaker saw the difference in current between hot and neutral generated by this and tripped, just like it was designed to.
This also means you don't need to connect a GFCI receptacle to that circuit -- it's silly and redundant at that point, and also leads to the coordination problems you're seeing where two GFCIs are tripping in reaction to the exact same ground fault. Thus, the easiest way to solve this is to replace the GFCI receptacle on that circuit with a regular receptacle with a "GFCI Protected" label on its front and let the GFCI breaker do its job.
(P.S. a more reliable test of a GFCI is to use the "test" button on the unit -- that shunts current from load-side-hot to line-side-neutral, which will also trip the GFCI, but does it independent of any problems with the equipment ground to the outlet.)
Grounded circuit (green/bare ground wire wired properly):
- Push "Test" on the GFCI protective device, and it trips -> PASS
- Push "Test" on the GFCI protective device, and it does not trip -> FAIL
- Push "Test" on a plug-in GFCI tester, and it trips -> PASS
- Push "Test" on a plug-in GFCI tester, and it does not trip -> FAIL
A fail here indicates the GFCI unit is probably defective.
Ungrounded circuit (green/bare absent or defective):
- Push "Test" on the GFCI protective device, and it trips -> PASS
- Push "Test" on the GFCI protective device, and it does not trip -> FAIL
- Push "Test" on a plug-in GFCI tester, and it trips -> FAIL FAIL FAIL
- Push "Test" on a plug-in GFCI tester at the GFCI receptacle, and it does not trip -> PASS
- Push "Test" on a plug-in GFCI tester at a downstream protected outlet, and it does not trip -> MEANINGLESS
Fail at the GFCI device probably indicates it is defective.
Fail on the plug-in GFCI tester (i.e. it trips!) indicates they have bootlegged ground at the GFCI receptacle- attached the neutral wire to both neutral and ground. It will seem to work at the GFCI, but is still dangerous.
Bootlegging ground at a downstream GFCI receptacle is a mistake, because one of several electrical faults could put 120V on the the grounds, e.g. the cover plate screws or a machine chassis. However this is difficult to detect, since a properly wired downstream receptacle will behave exactly the same way. This means for ungrounded downstream receptacles, plug-in tester testing is completely meaningless.
Once you have settled the question of bootlegged grounds, here's how you test an ungrounded GFCI. Plug your GFCI tester into your handy dandy 2-3 prong "cheater" - the kind with a short green wire as a pigtail. Extend that green wire all the way to a reliable ground source, e.g. the panel in the basement. Now, the GFCI tester should work normally, since you have rigged a proper ground to it.
GFCI protection is pretty effective, and I would be confident in an ungrounded circuit if it has GFCI protection. However if you are unable to get the external device to trip, you'll need to pop the cover off and see if the ground is present, missing or bootlegged.
Best Answer
If the receptacle is grounded: Yes it should trip it.
If the receptacle has some semi-grounding that is insufficient to clear a dead-short bolted fault (i.e. incapable of flowing 200A without setting the house on fire): Yes it should trip it. And that is fine. A bolted fault is also a ground fault, which should trip the GFCI, ending the event.
If the receptacle is not grounded: No, it should not trip it. That would be disturbing if it did.
If the receptacle is not grounded, but a tester indicates grounded: Then a) no, it should not trip it, and b) that is a bootleg ground that defeats the entire purpose of even having a GFCI /facepalm /Darwin_Award and needs to be corrected ASAP.
If you stick the GFCI tester in a 2-prong cheater and come off the ground tab with a separate wire run across the house to a part of the electrical grounding system: then yes, it should trip it.