Part of the problem here is the sensitivity of the inverter's built-in self-protection circuitry. Some respond to low voltage or high amps extremely quickly and may shut down with just a few milliseconds of high load. There is some good reason for that - the inverter output voltage and frequency is highly dependent on the DC input voltage. If the battery voltage sags badly for an instant, the 120v & 60 cycle output may fall outside the acceptable range. Since 60 cycle alternating current peaks in voltage every 16.67 ms, the time window is short unless the inverter circuitry does some averaging to assess power over a somewhat longer time period. Still, it is going to be a very short time as best. Therefore extra battery capacity (AH) and short, large gauge, cables are key to keeping the inverter happy during start-up loads.