Hmmm. Curiouser and curiouser. Rapid clicking noise could indicate low battery voltage. Climate control closes a relay (click) but the battery supplying it is too low or corrosion of wiring terminal(s) is causing a high resistance (unclick), which decreases the load so the voltage goes back up (click) - back and forth.
And you have problems with not getting AC shore power. Which means no AC power to keep the batteries charged up so they slowly fade away. Except you told us you plugged the converter into shore power separately. GAH!
Yeah, hire a mobile tech...