Hello Andy, I know you said you tried a new LCD, inverter board, and inverter cable. Have you tried a new Display Data Cable? MacBook Display Data Cable If the backlight is going in and out when your moving the screen it could be this cable.
It involves a little more work, but you could also try lifting the logic board off from the battery just enough to let the phone reset everything. Then putting the phone back together and turing it on with the display in.
Try going into system preferences, and then clicking on the Energy Saver icon (looks like a light bulb). There should be 2 tabs, one saying battery and the other saying Power Adapter. Towards the bottom left of the window, both should have a check box that says (Show battery status in the menu bar. Check the box and see if that fixes it.