> I guess you'd notice a change in light frequency based on the light in front/behind. Redder behind, bluer in front.
The light in question is not uniform like white noise; the spectral power distribution has relatively light and dark lines in them as a result of the physics of the bright sources and intervening gas and dust. Those features also get redshifted.
If one is moving relative to the sun, one would pay attention to the sun's Fraunhofer lines https://physics.weber.edu/palen/clearinghouse/labs/Solarspec...>, which would be Doppler shifted to different wavelengths. These lines also appear in reflected light from bodies in the solar system; if you were flying towards Pluto you would see a corresponding blueshift of the reflected Fraunhofer lines (plus some additional structure related to the chemistry of Pluto; it has some luminescence, as does our moon, as do the leaves of plants, and luminescence tends to impinge on the narrower Fraunhofer lines).
Indeed, measuring the Doppler shifts of multiple known-chemistry light sources is a useful technique in navigation of spacecraft within our solar system; it can in principle do better than precision measurement of angles to multiple light sources.
The spectral distortions of the CMB are certainly interesting, but it's hard to imagine their utility for spacecraft navigation within the Milky Way, rather than helping to physical cosmologists understand why there even is a Milky Way.
In the solar system we have kind-of the opposite problem: in order to get reliable anisotropy data of the Milky Way, probes like WMAP need excellent almanac data for the ephemeris of Jupiter (it's a bright reflector of sunlight and its cloud-tops at ~70 kPa are ~22 GHz microwave-bright; I gather other outer planets are used too, but the details are beyond me) to check its 22-GHz-band detection of the CMB Doppler shift in the directions it looks.