Hi,
Don't want to waste your time, obviously... Would a contact microphone, the way a stethoscope is used, placed on each inductor one by one help to locate the clicking one? It might show a definite pulse that matches the timing of the click which is stronger than when placed on other parts of board? Or who knows, crazy idea - an MLCC on flexible pcb and using piezoelectric effect, detect pulse by change in MLCC change in voltage? Other naïve, newbie suggestion is worse and probably wrong understanding - mini inductor (with capacitor, maybe) waved over each inductor and used to couple signal, or just a THT capacitor with one end connected to a resistor, other lead touching each inductor top in case it senses pulse and generates a voltage...? Shame an ordinary LED (maybe in combination with other components) couldn't do something useful, like when placed under light bulb develops a Vf. Sorry for silly suggestions.