16 gauge should be okay if the batteries are reasonably close to being charged (and hence the converter isn't consuming a few hundred watts). Do double-check that the electric water heater is turned off (if you have one), and avoid using other high-draw items like the microwave or air conditioner or an electric space heater or an electric fireplace or....
I think it's worthwhile to buy a 10 or 12 gauge extension cord for the motorhome; there are times when you may want or need it when camping because the power available is not next to where you wish to park. It can also be handy for operating whatever devices might need operating from the exterior outlet of the RV.