Have you tried timing the "rapid ticking noise" to your heart rate/pulse after exercise? If it matches the heart rate then it's unlikely to come from your throat. I am not sure how the pneumonia episode several months ago could have caused this -- the only thing that comes to mind so far is an arrhythmia which occurs with exercises, triggered by recent infection. An exercise stress echocardiogram could rule this out (there is EKG monitoring during the entire test). The echo portion would rule out valvular heart disease.
I have mild asthma, and there are times when, if I bend over in the way you describe, I can hear a faint, rapid ticking noise in my chest. However, it is not in time with my heartbeat, and it mostly occurs towards the end of a long exhalation. In my case, it seems to be specifically related to airway irritation. I wonder if you are experiencing something related to your bout of pneumonia.
As the doc says, see someone and get it checked out just to be on the safe side.