Really? These structure constants never involve any units of distance? Isn't distance changing at all levels?
If I have a meter stick, and then space expands by 10%, you're saying that my meter stick will end up being 0.91 new-meters long, yes? Wouldn't that require the structure constants to change, to allow for the same molecules to take up less space?
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.
The meter-long stick you put into that space to measure it is still one meter long at some time in the future because the force that pulls the space apart (dark energy) doesn't pull on the matter that makes up the meter stick strong enough to overcome the forces that hold the stick together. But even if it did, the idea of one meter hasn't changed its definition. It's just the tool you used to measure one meter has broken.