Sure; see here:
https://en.wikipedia.org/wiki/Relativistic_Doppler_effect#Doppler_effect_on_intensity
The derivation is elementary and can probably be found in many textbooks.It's not "redshift alone". It's "redshift in any spacetime geometry". Or, more generally, "redshift due to any state of motion in any spacetime geometry".
For example, the "expansion of space" derivations you quote are for comoving observers in FRW spacetime. But we on Earth are not comoving observers; we see a dipole anisotropy in the CMB, whose magnitude and direction changes over the course of a year as the Earth orbits the Sun. In other words, relative to a comoving observer in Earth's vicinity, we see a redshifted CMB in some parts of our sky, and a blueshifted CMB in other parts. And yet we observe the CMB to be a black body in all parts of the sky (just with its temperature redshifted or blueshifted from the nominal 2.7 K that is the "corrected" value after the dipole anisotropy is removed, i.e., the value that would be observed by a comoving observer in Earth's vicinity).