Let’s come at this from the other direction: if everything including the space between atoms and galaxies expands by 10%, how do you tell the difference between an old-meter and new-meter? For this to be observable, something has to still scale by the old-meter, or you wouldn’t be able to observe a difference at all. That something would be the physical constants with units, like the speed of light. This means the electrical force governing the chemical bonds in your meter stick will still try to restore the distance between them to their original old-meter distance, but in new-meter units.

Then if you define the actual meter by the length of your meter stick, or the physical constants, you end up getting the same meter that everything not macro-scale “snaps back” to.