Relationship between the magnetic dipole of the electron and the polarisation of its radiation

You're asking about the polarization of synchrotron radiation, which can be described to high accuracy using only classical electromagnetism. The polarization is due to the geometry of the charged particles' trajectories and has nothing to do with the electron's magnetic moment. Compare to the freshman-physics calculation of Brewster's angle, where strong plane polarization appears because the electric field can't be parallel to the reflected ray's direction of motion. If the Wikipedia page on synchrotron radiation isn't adequate for your purposes, I'm pretty sure it's the subject of a chapter in Jackson.

At a polarized-electron accelerator, like the one at JLab, the electron polarization precesses as the beam is transported through the machine. The amount of precession depends on the strength of the steering magnets but also on the beam energy when it interacts with those steering magnets. The beam is "born" polarized and can be delivered to an experiment with the polarization pointing in any direction on the sphere. Computing this polarization transport is something that the engineers teach the graduate students to do as part of making sure the polarization is what the experiments need.