Bryant,
I do not have an account on that forum so I can not see those diagrams but from what I can see it seems he has hacked the inverter internals and controls the internal charging relay directly.
As for inverter control I'm quite sure that remote ON/OFF work it just needs to be setup as switch type (not sure how that is done likely by setting a jumper). The inverter then should be OK and can be turned ON by shorting those two pins likely one of them provides the 12V or whatever voltage is needed for the second pin.
The CPC1705 can accept any voltage you want and all you need to do is limit the current to around 5 to 10mA using just a simple resistor calculated based on your available voltage can be 5 or 12 or even 24V
The CPC1219 is only capable of switching 200mA (may be insufficient to control a relay coil) and the just added a resistor internally on the input.
I will not recommend you modifying the inside of the inverter if you do not understand how the CPC SSR work and the required resistor to limit the current to an LED.