My take on the matter would be to include both styles of physics, basically in the same light as UnknownPlayer has described. Consider the non-newtonian method a "training" flight mode for fresh pilots and planet huggers, where the overall movement is controlled by a system of thrusters that continually adjust output to provide the most "aeroplane" movement possible. Of course, the extra power required by this system means that turn rates and maximum speeds will be a lot less, in the tens of metres/sec. At the flick of a switch, the active RCS is turned off, newtonian physics kick in to full effect, and the player can spin on a dime and accelerate to reletavistic speeds.
In practice, relativity wouldn't come into the equation in the case of a dogfight. Even if the spacecraft could accelerate at 9G continuously it would take months, if not years, to reach .5c where such distortions would begin to rise to prominence. While a fighter could attain some whacky speeds, going too fast means you overshoot targets and waste energy, meaning that most battles would still be conducted at a leisurely pace. However, there's nothing stopping some fictitious anomaly (or even the story fountain that is a black hole) introducing relativistic physics and situations into such a game. Maybe in a couple of years