I could be wrong, as I'm a bit rusty on this topic as well, but I was under the impression that the electric field was only distorted by relativistic effects if one of the electrons is actually accelerating. If it is just zipping along constantly at its high-fraction-of-c then I believe its corresponding field appears to be doing the same thing _for all observers within the radius of the radiated EM field that was created by the acceleration of the electron[1]_

Or have I made a relativity-noob mistake and what I said is only true in the frame of reference of the moving electron?

[1] Sorry for not having a better way to describe this concept. If it has a name (maybe light cone? Not sure) I'd love to know.