Well, pronounced doppler effects might be MORE realistic. The reason being:
I figure that since there is no sound in space, your fighter must be playing the audio for you, sort of like an in-cockpit surround system, in order for you to get a tactical advantage. It would turn its scanner/sensor findings into sounds that the pilot would hear and recognize. In the case of scanners failing, optical sensors would perform the task as well. I'm sure by the time we have giant lazor beams of death, we'll have computers capable of handling this. An increased doppler effect would make it easier to judge the velocity of sounds relative to you, increasing your tactical awareness.