I find the jump start issue to be very odd. If the batteries are new and charged there should be no benefit for jump starting. What are the voltage readings? Are you certain the new batteries are/were initially and still connected with correct polarity? As long as the battery has 10+ volts it should work fine without a jump.
Tripp-Lite is known for being a superior unit.
This is model RV1012ULHW yes?
https://www.tripplite.com/1000w-powerverter-rv-inverter-charger-hardwire-input-output~RV1012ULHW/Have you checked the 12 amp input breaker?
Also disable the load sensing while you figure this out.