I agree with two of the posts above, that #10 wire between the inverter and the battery will work well, and there will be much less voltage drop to the inverter.
You seem to rarely need the inverter, so a larger one seems not required. Either way, you probably would need #8 wire for the larger inverter, and the small one will barely have any voltage drop across the really large wire.
Also make sure that the return wire is also copper wire to the battery, so that you are not using the steel frame to conduct power back to the battery.
Check the battery for loose connections. A loose connection such as the battery ground can cause momentary voltage loss that you described.. .
By the way, a larger inverter would also require dual batteries. Dual batteries, and your voltage drop would be much less, so not requiring the replacement inverter.
Also a solar panel can allow the battery to be fully charged and be in much better condition before a trip starts.
SunElec.com has a 140 watt solar panel for about $229 that has a aluminum frame. It will work great on any RV!
GOod luck,
Have fun camping!
Fred.
