So is the clicking still happening? If you disconnected the house battery and the shore cable, there should be no power for it to 'click'.
Q1. Did it rain before you found this issue, that is did any part of the generator get wet?
Q2. Were you able to find the source of the 'clicks'? I can only think of the Onan starter relay, the electric fuel pump or the fuel cut-off solenoid... but may be others.
Q3. what model is this unit? None given
so I ass.u.me it is the Microquiet 4000 KY