The Sungold 4000K LFP inverter/Charger, I believe you are using looks almost identical to my AIMS 3000K inverter/Charger including the Manual.
I was able to open the case and solder parallel wires to the on/off switch and to the Rotary Battery Type Selector switch.
And give the SBMS0 complete control over the Charging and on/off functions for the inverter/Charger.
The post I wrote earlier on it:https://groups.google.com/g/electrodacus/c/jAwPWd9W4B8/m/pYlPamHqAQAJ
Peter