Nick, I could be wrong, but I suspect the heart of the problem is that you are dealing with two completely different sources of light. Reflected sunlight is illuminating everything in the picture except, you also have two bright point sources of incandescent, or LED, red light shining directly at the camera. Given the major difference in color temperature of these sources, even with a perfect exposure I don't think you can get the WB of both captured correctly by the camera in the same picture. If you really want the lights to be red, I think you'll have fix it in PP.
To check it out, try shooting one with the WB set to incandescent and see what traffic lights look like. Obviously that will mess everything else up, but you might get red lights.