A hydrogen atom 6 billion light years ago (when the universe had expanded less than now) has exactly the same mass as one now. Think mass in its comoving frame of reference which is locally inertial.
It emits the same frequency light as one here on Earth (ignoring our peculiar velocity). The frequency from 6 billion light years away gets redshifted by the expansion equation for redshift. Everything else you brought up is irrelevant. In that sense the mass of a hydrogen atom is conserved. The thing that looses the energy is the photon, and that goes into the expansion if you want to think in terms of energy conservation (that gravitational energy gets created in the expansion), or that the energy is lost if you want to follow strict relativity.
Added in response to Rankins's comments below
Yes to the second, no to the first. Here's why.
The first: why neither the Rydberg constant nor the electron mass change in a gravitational field? The electron mass is a property of the electron. It used to be called the rest mass. The rest mass enters in general relativity (GR) as an invariant, what can change is the energy and momentum. In a local frame, or an inertial frame which in GR is any freely falling (nothing but gravity) that is its mass. In the FLRW universes, for instance with $k=0$ (flat, but if not it's similar relations), $\rho a^{-3}$ is what does not change with the expansion, with a the universe scale factor. The density $\rho$ is mass per unit volume, and so mass stays invariant as the volume goes as $a^3$. The evolution of the universe when matter dominated goes like that. It matches all the cosmological observation and standard model confirmed by the latest CMB observations.
The energy does not change when the universe is matter dominated. But when radiation dominated (e.g. photons), it does loose energy in the redshift. It is simply from the equations for FLRW of the photons. When matter, energy and cosmological constant are all relevant each evolves its own way, mass is conserved, energy is not.
The spectra from stars and galaxies have shown no variation even from quasars and white dwarfs except for that accounted for by the redshift, with measurements of that possible variation to better than 1 part in $10^7$, even at redshift of 2-3 (or time back of 10-12 Gyr).
In special relativity we used to say the mass grows as the velocity gets closer to $c$. Nowadays one says in both SR and GR that the mass of an elementary particle is an invariant, the rest is momentum.
Now for the second, how or why mass energy is not conserved in GR. For energy you already know. For mass, it is for elementary particles, but for composite or macroscopicl bound bodies the mass can change because the binding energy, internal momentum, or simplistically gravitational potential energy, can contribute to the mass, and the energy in this effective view (the total mass or energy, ignoring the external momentum, of the composite body). Thus, when the two black holes merged in 2015 the sum of their masses was 3 solar masses less than the mass of the final black hole. The 3 solar masses were lost to gravitational radiation.
And in general, GR does not have a covariant measure of the energy of gravitational waves, or the total in energy in spacetime. For highly dynamics spacetimes or events it is not generally conserved. For asymptotically flat spacetimes you can get global (but not local) conservation, i.e., from the flux at infinity (or far enough away, like us and the black holes).
If you are not completely sure what is meant to be included in the energy and mass, such as in the simple cases I described above, you have to take the full stress energy tensor as the gravitational source, which includes energy, momentum, angular momentum, and stress.
