Is your inverter a whole house unit , or are only select 120 outlets powered?
Is the inverter a multi function ( inverter/converter /charger)?
Could you post the make and model of the inverter?
In a multi function unit there will be an internal logic module & transfer switch in the unit , that gets a signal from the on/off switch, and looks for 120 volt input from shore power or gen set ; then decides which function it should be in. ( inverter or converter/charger).
With the re-set ( re-start) method you describe ; involving the boost switch ; the inverter may be getting a weak signal ( voltage) from it's on/off switch, that requires the input from all batteries to overcome, and if it relies on that voltage reading ; on the wiring from the on/off switch; to determine the batteries state of charge . The unit may be wrongly thinking ( as Dr4Film mentioned) that the batteries are too weak to be inverting power from.
You problem could be as simple as a loose connection on the on/off switch for the inverter, or at the unit it's self.