Hard to follow your OP question.
Yes, if your inverter is connected to the battery (properly - pos and neg in the right spot) AND you turn the inverter on then the female inverter outlet should be giving 120 volts, as you confirmed by plugging a light directly into the inverter.
If the inverter is wired into the same wires as the outside connection to shore-power then the inverter should power up your entire panel and all the outlets in the camper, including fridge and Air Cond.
Inv will not run the AirCon, if Air Con is trying to come on then that is the first issue and could be shutting down the inverter, same for the fridge electric setting, same if you have a 2-way boiler set on electric. There should not be anything else hard-wired in that would make a big draw.
Re-confirm that the inverter is providing power as you could have blown a DC fuse that the inverter has on the backside.
Is there a reset switch on the inverter (built in circuit breaker)?
If you are handy with wiring you can wire it so you are not having to flip switches and remember what is plugged in and worry if the dogleg thing is going to back-feed your inverter and blow it out...
On my first rig I had shore power, generator and inverter all powering the camper.
Shore and gen went into the breaker box and that powered A/C, Fridge and Charger, and boiler. There was an A-B switch that only let one power source power the breaker box (gen or shorepower).
When I added the inverter all I was going to power was the outlets, not anything hard-wired into the camper, so made it easy.
I used a toggle switch of proper amperage to select power from the breaker box or power from the inverter.
I put this switch in-line with the wire coming out of the breaker box that fed the outlets. This isolated the inverter from the breaker box (a/c, fridge, charger, boiler), and let me power the outlets.
This made it impossible for the inverter to back-feed the breaker box or the generator or shore-line city power, which means the inverter would net get fed shore or generator power either.
I wired the toggle switch from the "hot" wires so was choosing hot from either inverter or the panel. The ground and other wire, which was grounded, was shared.
Hope this helps you or anyone