the thought occurs to me that a high impedance 12v supply would have a better chance of starting the motor, if the UVLO don't turn off the inverter. the reduced voltage would cause a likewise reduction of output V and I. many of these 12 volt inverters turn off at 10.6-8, to protect the lead acid.
if you can find the output current transformer, you can increase the typically 10uF cap after the rectifier to 50 or more. if the inverter measures current in real time, you can't do much, especially if it is uProc controlled.
the 12v side fets won't mind the extra current, provided normal operating temp isn't extremely high....