GFI Leviton LB115-DS Testing at outlet with GCFI Tester
If this were a old home I would say no problem if the tester on the breaker worked. But since this is a new home get the contractor to connect the grounds.
The GFCI can and will function correctly as it doesn’t use the ground at all in its circuitry. The GFCI circuits detect a difference in the current on the hot and neutral if there is an imbalance they will trip.
Your circuit tester places a resistance of ~15k from the ground to the hot causing a properly wired receptacle on a GFCI breaker to trip.
If both your’s and the inspectors testers did not work some one was lazy and did not connect the ground and that is why the GFCI test is not tripping the circuit.
It can be a simple case of a wire coming loose but make the contractor fix it it will probably be a simple fix.