I think the "click" is generated electronically by the signal that feeds those green turn signal lights. It is possible that changing that indicator will make current flow and generate the 'click'.
It is also possible that the connection form the console switch group to the instrument panel is bad. ON the back of the instrument panel, check for pulsing voltage at C1 pin 1A with the turn signal activated.
If you see no pulse, the issue is in the console switch group or harness. If you see the pulse, the problem is in the Instrument panel.