To have shorted input would mean the inverter couldn't function at all. Either never tested at the factory, or blown after test?
To have no continuity out of the box sounds like no decoupling capacitors connected directly to battery terminals. Is there a battery breaker in the inverter? Order of operation, as jyoung1 mentions.
So many people have reported inverters which ran just fine on every battery they tried, except the one they bought from SS. The inrush to a capacitor can be massive, trips BMS, blows class-T fuses. Yet none of us ever had that problem with lead-acid. We suspect there is variable behavior between inverters, even of same model, and marginal performance of the built-in precharge circuit.
Can you connect the "fault" inverter to the EG4 battery (or any battery, even 12V car battery or flashlight battery) through a resistor or light bulb and measure voltage? Curious whether it does charge up, proving "continuity" isn't a direct short.