Does the Amazon Basics 500W Heater work on an 800W Inverter and 12V car battery? 

A 12-volt battery is normally attached to an alternator

Something else you can try is to run a 12V heater built with Peltier elements. That way you eliminate the losses into you inverter.

It'll work, but you have to have the engine running.

What if you had it in a vehicle and the vehicle was running. Would it be able to run the heater then.

the case is made from metal or hard plastic?

Why don't you run your test on a fully charged battery?

Depending on the battery you had you can expect different run times. Because your inverter has an efficiency of about 80% we can estimate you will draw about 600W out of the battery, meaning you draw about 50 A. A regular car battery should read 12.8V when fully charged and it will have a capacity of about 70 Ah. That capacity is calculated like 7 A drawn for 10 hours. When you draw 50 A you can expect about one hour of run time. Try again with a fully charged battery.