I just observe that a lot of cosmology seems to be riding on the theory that the red shift is caused by an expanding universe.

This seems wrong to be. There's at least two independent lines of evidence for the Big Bang theory besides redshifts -- isotope abundances (particularly for light elements) and the cosmic background radiation.

What if it light just loses energy as it travels, so that the frequency shifts lower?

We would have to abandon our belief in energy conservation. And we would then wonder why energy seems to be conserved exactly in every interaction we can see. Also we would wonder why we see spontaneous redshifts not spontaneous blue shifts. Every known micro-scale physical process in the universe is reversible [1], and by the CPT theorem, we expect this to be true always. A lot would have to be wrong with our notions of physics to have light "just lose energy."

That seems like a perfectly natural solution. How do we know it isn't true?

This solution requires light from distant galaxies to behave in ways totally different from every other physical process we know about -- including physical processes in distant galaxies. It seems unnatural to say "the redshift is explained by a totally new physical process, and this process violates a lot of natural laws that hold everywhere else."

[1] I should say, reversible assuming you also flip the charges and parities. That's irrelevant here, though, since photons are uncharged and don't have any special polarization.