I'm pretty sure we were talking about the ensemble of redshifts across the sky, rather than one particular photon. (Even in the single-photon case there will be lots of uncertainty about the \lambda_{obsv} vs \lambda_{emit} relationship; think in particular about its interaction with ions along the way. We'd want at least an "ensemble" of photons along the lines of a classical beam equipped with a spectrum).

I'm not sure if you are objecting to approaches like the tetrad formalism in your final sentence, or even to gauge freedom itself. There's no reason an observer -- even an always lightlike one -- can't carry around its own system of coordinates. People from ISS astronauts (LVLH vs ECEF/ITRF) to aerobatic pilots (LVLH/LENU to GPS) to pedestrians looking at map apps on their smart phones do that casually all the time, and nobody expects ensembles of objects to move with respect to each other as one swaps sets of coordinates. Setting a new coordinate system every few steps while navigating on foot through a complicated city core is perfectly fine, and so is setting up a new coordinate frame at every infinitesimal step along an affine-parametrized curve traced out by a photon. IMHO that is the core of relativity.

In arbitrary spacetimes, redshift has to be computed along specific worldlines, ie you tend to treat photons as point particles. I guess in principle, you could solve the Einstein-Maxwell equations (or more likely, the Maxwell equations on a fixed background), but I wouldn't recommend it. There are special cases which are simpler to handle (eg if your spacetime is static and you use adapted coordinates, the time coordinate will be cyclic in the particle Lagrangian, the canonically conjugate energy will be conserved and redshift will be given by the quotient of the roots of the g_{00} components, which formally comes in via normalizing emitter and observer 4-velocities).

I'm not sure if you are objecting to approaches like the tetrad formalism in your final sentence

The final sentence was just a comment on the pedagogy of deriving cosmological redshift via infinitesimal Doppler shifts. There's nothing wrong with it (sections of the frame bundles and all that jazz if you want to get formal), but to me, this feels like putting things on their head: I consider the Doppler shift a special case of the generic relativistic shift, making it circular to then explain the generic case as a series of infinitesimal Doppler shifts integrated along a photon's worldline.