very few people connect an inverter to the converter battery charge line, because it is usually 'Under Sized' for large inverters, and a long run from the batteries
yes..mount inside up front, and go thru the wall to the batteries
the alternate is mount near the batteries in an outside compartment, and use a 15 to 30RV pigtail and plug the shore cord into the inverter, this powers all the TT 110v outlets, no special inside dedicated wire run needed
finally as noted, mount inverter near the converter, attach inverter wires to converter battery wires, route inverter 120v power to outlets using a transfer switch, break out the outlet feed from the CB, and run thru the transfer switch
i think either method one or method two, are your best choices
method three offers the chance to mis-wire something at the TS
