Your wiring is apparently not actually what you think it is. I'd double-check the connections to the inverter and at the circuit breakers as what you describe should not be happening. (I'm assuming the two breakers are both on the positive leads and not the ground leads. If they're on the ground side, then it's nearly inevitable that there would be cross-paths between the two systems, among other difficulties.)
Not that it matters for your situation, but why only a 30A breaker for the house circuits when the wire to the (assumed) house distribution panel is 4 gauge? I guess if you never exceed 30A it's a moot question, but I'd think 50A or more would be handier.
You do have your inverter connected through the tri-metric shunt, so it would show the current flowing through the inverter. Somewhere around 55A on the 12V side is just about what one would expect for your 5.6A 120VAC load. There's nothing strange going on there. (And I think you do want the inverter power to be measured by the tri-metric so it can give reasonable readings for your battery states.)
