The current version of the board does not get ground through the screws. It gets ground from the 3 pin screw terminal, see the diagram below:
There are 3 things that can cause a false no ground warning when the service ground is known good.
1. The ground connection to the OpenEVSE board is not securely connected with the AC ground and the EV Ground.
2. The AC_Test lines are not securely connected to the EV (Load) side of the relay.
3. The relay is not closing properly.
Ground is checked 1 second after the relay closes. With the test disabled the controller is assuming the relay has closed as commanded, which is not the case here.
If your relay is 12V... Your 12V Relay coil should connect to Relay 1/2 Common which is an always on +12v. The other end to either Relay 1 or Relay 2, these are 2 independent channels which switch in a ground to complete the circuit. If It is not working on Relay 1, give Relay 2 a try.