A second battery with battery isolator.
If you have a charge lamp, try the "UIBI" that I've described on mp3car (google "mp3car izu069 uibi).
PS - the cig socket powers off during cranking.
If your battery is strong, you could try direct wiring to the battery thru a relay that is powered by IGN +12V (which stays on during cranking) or a manual switch.
But an auto-isolated 2nd battery is the best option, especially if there's a chance the inverter will flatten your main battery, or if excess cranking, cold weather, or age dips the cranking voltage below the inverter's shut-down threshold.
If you have had an over-voltage shutdown, check your charging voltage, though it may be a short voltage surge that is triggering the shutdown.