Update: So far so good, a few adjustments needed.
So a couple of my eggs started to dent a little, requiring the addition of some damp toilet paper (we were out of paper towels). A small, crumpled up piece in the center of the circle seemed to do the trick nicely. No need to actually cover the eggs, which makes me happy because I can still see them without opening the container.
I also noticed yesterday that I'm getting some mildew on the sides of my inner container and that an egg that was touching the side was also getting some mildew. I moved the eggs over just a little so that none is touching the sides and today the mildew on that one egg has decreased. Good. I think today I'll just change out the container. In this set-up, that's super easy. I'll just pick them all up, plop them in a different container, and plop the new container back into the bain-marie. Easy as pie.