Figured this out on my own. Each object needs to be given a reference to an Array containing references to the FlxCamera objects you want it shown on, and this needs to happen (usually) when the object is first made. The first time a FlxObject calls update(), if its cameras is null, it assigns FlxG.cameras as a default, which means all of the active cameras will display the object.
I did this by making a few static Array's in my Main class, one for each camera group, and then in the constructor for my various classes, I would set their cameras variable to point to the corresponding Array.
The biggest frustration: Currently FlxGroup does not pass its cameras on to its members. Hopefully this will be added into future versions of Flixel so that FlxGroup's can be assigned a camera group and have all their children also automatically assigned the same camera group.