why not just switch the mains coming into the inverter to start battery charging by the inverter? That's how I have mine setup. Currently I have Pb batteries and they don't like sitting discharged(real bad for them). I use a aux relay in my XW60 charge controller as a low voltage switch to a HVAC relay. Keeps the batteries charged above the low voltage dropout setting on the inverter. I went with the KISS theory.
Read this:
"In its most basic form, a transfer switch for a 120/240V single-phase service is simply a double-pole, double-throw switch of proper ampere rating with a middle off position (Fig. 1). By design, it is a physical impossibility for both power sources to supply the load simultaneously, unless, of course, that is the intent. It's also impossible for the two sources to mix or for one source to backfeed into the other."
Your one line shows 2 contactors which if only one changes state could cause issues to 220v loads if the phases were not 180 degrees from each other.
