The link you provided doesn't shed much light on the operation of this inverter, so I'd assume that it uses a bit less than half of its stated output to idle, (approximately 100w), which would take 40 - 60 hours to flatten your car's 12v AGM battery.
Of course, you'd be using the car some of the time, and this would mean that it could be left on indefinitely, providing you didn't run the battery down to less than half charge, (perhaps 20 hours, depending on the load applied to your inverter)
Caution is always the best approach, and small inverters do tend towards optimism, so perhaps a bigger one?
Try to investigate the overhead specification that the inverter has, a better one may not boast as much, might do a better and more efficient job.
Consider also, the age of the AGM battery, it could be getting ready for replacement, and this exercise might help reveal that, (this might be good or bad, depending on how you look at it)
My experience with a high quality 100w pure sine wave inverter is limited and recent.
I'm able to pull 800-900w before the unit starts complaining, so it is doing what it is advertised for.
A cheap one might not cope so well.
The overhead I've got to contend with is about 6%, so idling uses about 60w. It comes with a remote control and a long cable, so this isn't likely to be a problem.
A small 12v portable battery would be good to keep handy in case you overdo it, then you can jump start your car with it.
Using an inverter connected to the car's battery while it is not charging or running, isn't recommended.
The car's AGM is charged anytime the car is plugged in or, 'ignition', power is on.