Sorry about your dilemma. It occurred to me that it might be fear of that unknown thing that is getting to him. Can you open things up enough for him to see that there are no bogey dogs under the settee when the A/C kicks in? Maybe if he sees that the clicking is just coming from yet another mysterious machine and when you demonstrate that it is nothing to worry about (by petting the A/C when it clicks? jajaja) maybe it will sink in that there is nothing really for him to worry about. Something to try at least...
BTW, I had never heard of "anxiety vests" before I saw your mention of the Thunder Shirt. I have had dogs that were absolutely terrified of thunder while another sleeping right next to them doesn't even raise an eyelid to see what the fuss is about. Now I know what to get one of my daughters for Christmas! Thanks!
~~ Red