I don't think there is a way to get the AABB for an Area in Godot, at least not by default. You can make your own AABB manually through code though, as long as you know the rough size of the shape and its position.
Another way you might be able to achieve the same effect, though it's more complicated and computationally costly, is having small Area nodes on each corner of the Area. Then you can use these nodes to detect if the object is fully enclosed in the inventory by checking to see if each of the Area nodes on the corner are inside the same collision shape. I'm fairly positive this would work, but it seems rather costly.
