This is hard to explain in a few words, but I will try. I am concerned that the bare neutral conductor is acting like a neutral grounding electrode for the whole house and seeing fluctuating current to ground caused by changing loads in the house. Since the EVSE checks for a safety ground connection by injecting a small current from line to ground, fluctuating current in the bare conductor might cause the EVSE to shutdown. This would be less likely to affect the 120-volt EVSE because the ground sense current is from only one line to ground and the 12 amperes of continuous return (neutral) current on the bare conductor would tend to dampen the fluctuations. I have seen severe unbalance between the two 120-volt lines and damage to metal water pipes from inappropriate ground currents caused by loose or inadequate neutral and ground connections. As others have already noted, the feeder to the garage should have 3 insulated conductors (L1, L2, N) and a dedicated ground conductor.
Silver LEAF 2011 SL rear ended (totaled) by in-attentive driver 1/4/2015 at 50,422 miles
Silver LEAF 2015 SL purchased 2/7/2015