So, I abandoned the square cavern geomorphs. My tests just didn't work out. Over on the DM's Craft board, a person named @onethatwas suggested hexes.
This idea worked better. I had a few quality control issues making sure that the hexes were exactly the same size, but they'll work fine.
Here are some pics --
The hexes are double-thick cardboard and the walls are a single layer. The "door" gap is 2" and the hexes are 10" across. I spray-painted them black and then I drybrushed the edges with grey. I also used a sponge to highlight the "walls." In some spots (you can see it on the tile above) I used the sponge to make slime/mold/fungus on the tiles. Just for visual interest.
At this scale, they will integrate with my Dwarven Forge Game Tiles, so we'll see how that goes.