As I've mentioned in another post, that leads to questioning one of the most well tested theories in physics, so you need extraordinary evidence to prove it over something as elementary as doppler shift. Like, if it's Earth's gravity well causing us to see things differently, then things that are further from Earth should observe things differently.
There is already extraordinary evidence that something in physics is behaving differently at larger scales: the behavior of galaxies (spin, gravity pull) doesn't match their mass. Any mass. There is no single mass value that predicts their entire behavior correctly. Thus dark matter, dark energy etc competing theories, which are so far untestable.
It wouldn't surprise me if we discover someday another Doppler mechanism that occurs at those same large scales
