The small relay is used for the Ground Fault prove out. This simulates an unbalanced current situation which tests the CT and GF circuitry. The failure mode for most GF circuits is the CT or the buffer/detect circuitry, so without prove-out, you would be risking the dangerous situation where the GF detection could fail and still allow the EVSE to energize the load.
They also perform a relay "stuck" test to detect if one or the other relays have welded contacts, and if so, the opposing relay will be de-energized. The ground verification test passes a small current through ground to verify a low impedance path.
This is why all your simple bypasses and re-routes failed, and if you bypass or fake any of the prove-outs, you are creating a dangerous situation. So again I will stress that you should not explore this path.
You will find, as I did, there is no $20 path to convert this EVSE to 240v operation, let alone a SAFE method. It can be done relatively easily with the right methods, but by the time you add up all the costs, you will find it well outside of your $20.
I spent several weeks reverse-engineering the circuitry and firmware so I could write a new version to enable the 16A pilot. Any other way will require addition of a lot of extra circuitry most likely including another microcontroller. By the time you do all this you will probably have spent 100 man-hours and at least $50 in parts. In any event, not an upgrade recommended for the hobbyist.
Then if you hope to sell your upgrades, by the time you pay your insurance, answer all the emails and phones, and all the other overhead, you will find there is not much to be made. I hope to be out of the upgrade business as soon as there are enough EV's on the road to warrant a business case to make a better EVSE for lower cost. The main thing I've gained from EVSEupgrade is less sleep.
I've been supporting the EV movement for almost 20 years, and I saw the availability of a low-cost solution for charging the LEAF as a stumbling block that may impact it's success, so this was my motivation for going down the upgrade path, something normally considered silly. I thought about "open-sourcing" my design, but if just one person got hurt because someone did something wrong, it would all be for nothing, and I'm not sure I'd want to live with that. If you are competent enough to modify your own EVSE, then it's pretty obvious how to do it, but I am not going to be irresponsible and release details to the general public on how to do so which could easily encourage someone to do something dangerous.
Keep in mind if you encourage people to do just that and someone gets hurt or killed attempting to follow your instructions, YOU could be sued and held liable for the death or injury.