@Keanen - Nice work !
It seems you have definitively identified the culprit.
I had assumed that these NACS to J1772 adapters were simply passive devices (containing no electronics, and simply making connections between the appropriate pins).
Comparing the two:
You can see that the signals correspond directly. The L1 and L2 lines have to be simply connected through. The Ground is straight forward. The only signals that I would question would be the CP and PP pins. These (especially CP) require a pretty specific voltage range / protocol with J1772, Perhaps the 'adapter needs to include some circuitry which 'translates' between the two rather than just connect them. Obviously Lectron did it right and Tesla did not !!
Both NACS and J1772 are supposed to use IEC
61851 which seems to imply that a direct connection of CP is possible.
The PP signal is related to the latch button and it informs the EVSE that the plug is present.
I am wondering this - Suppose the PP is at fault and the vehicle is not recognizing that it is 'connected' with the Tesla adapter. What if you plugged in using the Lectron adapter, BUT just stand there and hold the button in as though you were going to unplug it. Would that cause the same periodic clicking as it does with the bad Tesla adapter? That might further pin down whether the PP signal is responsible.
It would be very interesting to go at these adapters with an ohmmeter, checking continuity to see which pins just connect through and which do not. If you look at the circuitry associated with J1772, you will see that the PP signal does not even go to the EVSE. It goes only from the latch button into the vehicle:
The latch button is supposed to present the vehicle with either 150 ohms or 480 ohms to ground depending on whether the latch is depressed or not. If you have an ohmmeter you could try this on both adapters (for safety, don't try it while the adapter is connected to the Wall Connector).