I know I'm a few days late but just a few thoughts on this:
The purpose of a false bottom is to allow excess water to drain away to stop mould build up. False bottoms can be made by using the eggcrate method you mention or
by adding a layer of something called hydroton
to the bottom of your planted area then covering it with mesh designed to keep weeds out of flower beds. The hydroton
creates a water reservoir, the mesh
stops the substrate getting into the water reservoir thus allowing the water to climb and cause mould.
Hyrdoton is a type of clay that is used in hydroponics, it's great for false bottoms (I've used it before in a terrarium) and is typically available in large bags at a decent price. If you google "terrarium false bottoms" you might find a variety of articles about hydroton and its uses.
Even if you cut out sections in a foam background for your plants, the excess water would need to go somewhere that wouldn't interferre with the aquarium section of the paludarium; you could always rig up a piece of aquarium tubing in the bottom of your false bottom and then get a syringe (minus the needle) that fits on the end, you'll then be able to suck the water out of the reservoir if needed. In my own terrarium I didn't bother to put anything in the water reservoir as, to be frank, I didn't think about it. I didn't have any issues but I tore the terrarium down pretty quickly due to other reasons.