To minimize their time/your $$$:
With shore power disconnected, generator off and inverter/charger off:
Remove lid to ATS.
Tighten all connections and check for any burned wires.
Start the generator. Verify that the two on-generator breakers have been turned OFF and back ON.
Pull out a voltmeter and check for power on the IN and OUT side of the ATS. That will tell you if the problem is before the ATS or in the ATS itself.
10 minute job.