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
1. As other commenter said, X causes Y does not mean that Y implies X. There can be another cause for the Doppler.
And surprisingly,
2. There is at least one known mechanism that cause Doppler WITHOUT moving: when the observer is in a gravity well (ex: earth) and observing a stationary object outside the gravity well (ex: some fixed point in outer space)
It wouldn't surprise me if we discover someday another Doppler mechanism that occurs at those same large scales