I agree that the constants might be related in ways we don't know, which would allow compression. I'm more interested in an upper bound on the complexity than an exact value (which is likely incomputable for halting problem reasons), so I'm willing to be over by 100 bits because we fail to see a pattern.

As far variable constants: Sure, we can estimate the kolmogorov complexity where the "constants" are inputs. Or we can estimate the kolmogorov complexity of the current laws plus the state 13.7 billion years ago at the big bang. Or we can estimate the complexity of a program that runs all programs. All of these questions are interesting. But right now I want the answer to the one I asked, not the others.

edit clarified response