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Sure, but you cannot do 'cosmological Doppler shifts' in pure FRLW coordinates, because fundamental observers are not moving relative to each other - they are at fixed spatial coordinates, with zero coordinate velocity. Also sure, there are no gravitational potential gradients in FRLW coordinates, hence also no gravitational redshifts. To do cosmological Doppler shifts, you have to work in Schwarzschild (or equivalent) coordinates, including both types of time dilation factor, as applicable. Minkowski is just a special case of Schwarzschild. This is essentially what B&H, Peacock and others did.One point that the physics literature seems to agree upon without controversy is that that the clocks of fundamental comovers run at the same rate in FRW coordinates. E.g. Peebles textbook p. 59: ...

If you look at my prior post, you will notice that I did not miss that point. However, contrary to what you state, Peacock's equation 16 is valid for any matter-only universe, flat or not. Velocity- and gravitational time dilation will not cancel out for open or closed models, so the equation will give a different result to your 1+z = 1+v/c. I'm still busy testing eq. 16 in a numerical integration, so I'll comment more fully on it (and the rest of your post) later.Second, you seem to be missing the point of his eq 16: The velocity shift is a redshift, and the gravitational shift is a blueshift, so the two factors tend to negate each other. Indeed, his equation can be valid only if the SR time dilation and gravitational time dilation are exactly equal with opposite signs, which results in the emitter and observer sharing the same cosmological time.