Can an "absolute" frame of reference be determined by measuring the compression of light?

By compression of light you mean the Doppler shift?
Then yes you can measure your speed relative to the light source by comparing the Doppler shift in different directions.

It's been used for a number of different radio positioning systems - but it only gives you a motion relative to the light sources

Technically, doppler shift for light happens due to relativistic time dilation, so it is subtly different than acoustic doppler shift. It cannot therefore be used to determine an absolute reference frame.

That doesn't mean there isn't one. Spacetime as we know it is formed by the big bang, so that might be an absolute reference frame if only we can figure out how to detect it. Unfortunately, we can't, and in a very real sense the Big Bang happened EVERYWHERE, since all of space was compressed to a tiny point.