A converter would output DC, an inverter outputs AC power.Just as long as you turn on the inverter with the load connected, the 12V output will not be an issue. That 12V will immediately drop lower due to the load.
If you are idling a vehicle with a 140A alternator like most Siennas have, you will have 945W of power so with a 1200W power draw the alternator will be fighting a losing battle, using more power than you are generating and you will drain your battery (some Siennas have 100W alternators, check Auto Zone!)
The easiest thing if you don't want to install a backup battery system is more powerful alternator, you also need to add few thicker gauge wires. Or even if you are installing a solar power station with lithium batteries, a more powerful alternator will support recharging your system a LOT faster, especially if you are idling.