If fundamental constants could change, this would violate energy conservation, and the second law of thermodynamics. Someone once said, if your pet theory violates the second law, there is no hope. Or am I missing something?
Also, the second law is only applicable to closed systems. The universe may not be a closed system in the way we normally think of it.
My college physics professor once said, "if in order to make progress we must leave reality, by all means let's leave reality." He also pointed to three red volumes on his shelf, and said those may interest you, and they did. (Richard Feynman)
Some things may seem incredibly constant, but have to be measured in such a ridiculous small or big (time) frame, that it's barely not measurable at all.