Battery will actually last much longer if it is kept 100% charged. To do this the switch must be on for the converter to float charge the battery. If the battery is allowed to discharge it will slowly sulfate and lose capacity.
OK so the RV guy is partly correct from the old days of poor converters float charging at fairly high 13.8 volts causing excessive water use and risking damage from low water levels.
Best to evaluate your charging system when in extreme conditions such as 100+ or less than 30F. Typical converter still floats a bit high at 13.6 volts for 119 temperatures. If your voltmeter finds this true consider a replacement converter that drops to 13.2 volts for long term use or storage such as Progressive Dynamics or PowerMax.
Do check your battery water levels and add distilled water only if needed. Fill to 1/4 to 1/8 inch below the split ring that extends down into the cell. Just touching is absolute max. Use eye protection.
The 12 Volt Side of LifeAnd yes you need 12v power to operate many of the controls. If the converter fails the fridge and A/C will probably shut down and maybe more but with a connected battery all will be good for the day.