Panting is a normal way for keets (and for all mammals I think) to cool off if they get overheated. For example when we first got Ivory she had pretty much never left her cage before. So her wing muscles were extremely weak. If she just flew once or twice around the room she would land and hold her wings away from her body, panting. Both were ways to help her cool off again. I've had people write me whose keets were going crazy and panting, and it ended up they were overheating. When they put on the air conditioner, the keets calmed down.
So it could be nervousness or an extended fly-around that has your keet panting.
But as far as clicking goes, they don't have clickers in their throats
So I really think that must be the beak, unless what you're describing as a click is really a FWEEP or something else.
My keets' feet will be warm if they are moving around and get their blood going, again just like in most mammals. So that seems to reinforce the idea that your keet is somehow being active. Of course a vet visit is relatively cheap so if you're sure your keet isn't climbing around or flying to do this, it's worth a stop by the vet to see what else could be going on.