First you must assign BOTH the Ignition and the starter to a key or button. Starting the car then becomes a two step process. Press the ignition button, then press the starter button. (for me Ignition is a poorly worded choice, 'main power' or some other choice would reduce confusion. To an American ear, Ignition and Starter are synonymous. The key goes into the ignition. You turn the key/ignition which engages the starter.) So think of Ignition as the button that turns on all the electricals without actually starting the engine. Then pressing the starter button actually turns the engine over and starts it.
Now, here is one method to save time.
As you approach your pit box. Put the car into neutral or press the clutch pedal. While the car is coasting into your pit box. Shut off the main power switch so that your car is completely dead as you arrive at the pitbox. Once the fueling is done, and while the tires are being changed, you can power up the car but do not start it. Wait til the tires are done and then press the starter after being given the ok by your pit crew. By coasting in deadstick, your pitcrew does not have to wait for your engine to spin down. The ISI based code used by GTR2 does not instantly switch off the engine. It spins to start when your fire it up and then spins to a stop to shut off.
My biggest problem with refueling, in the early days, was the refueling sound takes several seconds to be audible. So I would shut down the engine, but because I didn't hear the fuel start flowing, I would assume the engine was still running. Thinking I needed to once again shut it down I would hurriedly press the starter button again. About that time, the fuel flowing sound fx would begin and suddenly stop because I had just restarted the engine and now the pitcrew would have to wait for me to first shut it off and then have the engine cease producing power completely. A real time killer.