It took billions of years of universe-stretching for the CMB to redshift to microwave, I don’t think me pushing pedal to the metal for a few seconds on a rocket ship is going to counteract all those years.
We do see shifts in the CMBR due to local velocity changes though—for instance, we can tell that the sun is moving at about 370 km/s relative to the CMBR frame due to its radial movement through the galaxy and the motion of the galaxy itself through space.