Quantum,
3 paralleled little 9V batteries are good for only about 60mA. Therefore their power output is only about 1/2 Watt. The inverter will have some loss, so its output power will be less. If you power it from a car battery, then its output voltage will be too high, but it will still be limited to the power ratings of its transformer and transistors.
What do you have that needs only about 1/3 Watt from 120VAC?
If you use a bigger, more powerful transformer and big, heatsinked multiple transistors, then you can get more power from it using a car battery to power it.