I have done exactly what you propose and it works great. Be sure to install circuit protection between the battery and inverter. I also installed a shunt meter, so I could monitor the DC power usage.
The one thing that you need to be aware of is the 12 volt charging voltage when the car is in READY mode. When the car is in READY mode, it will keep the 12V battery at 14.4VDC. This is fine for a traditional maintenance free 12V battery. However, while an AGM battery can be bulk charged at 14.4V, it should be float charged at no more than 13.8VDC. I have personally overheated my AGM battery while camping to the point that it vented. This damages the battery. Fortunately, I had already used the AGM battery for 5 years and soon replaced it with a traditional battery.
If you don't have JdeMO and you don't leave the car on for extended periods, the AGM battery is a good choice. The car does use a float charge voltage of 13.8VDC when the car is charging on J1772.