Best is to run the inv off the battery with the right size wire, and short wire runs.
Any inverter will work off the thin-lighting DC-wiring as long as you are not requiring much power from it. a 1,000 watt inverter will run a 10 watt thing and not blow the fuse, but ask for the full 1,000 watts and it will blow the fuse, or if no fuse, the wire will melt or catch on fire behind the wall.
Do not install a higher rated fuse if you are blowing fuses. Fuses make wiring dummy-proof in that your fuse will blow before you over-heat the wires.
Inverters will work best if the wire connected to the battery is able to carry easily all the Dc power that the inverter needs. Running power over a lesser/thinn wire will starve the inverter, heat the wire and cause loss of capacity. I am not sure that the inverter will be damaged, but I do know your Air Cond unit can be damaged from using too thin of wire over time (as an example of damage from low amps...).
