What you need to do is ground the "case" of the bulb holder ( Connect it to battery - ). One of the wires goes to the battery through an regular on/off switch and the other through the brake switch. Choose the wire that lights the bulb the dimmest for the running light. I wouldn't use a resistor to dim the light. You'll waste way too much power that way. Use a lower wattage bulb or LEDs to save power.
He will burn more power using just the bulb (15W). Using a resistor, he is only consuming 7W of power. Thereby, extending the charge of his battery. True, resistors are very inefficient, but you either loose the power with the resistor, the bulb or both. Perhaps you meant a resistor would wast too much power as opposed to using LEDs or lower wattage bulb? If that's the case, I agree (As long as the bulb is less than 7W)
But my concern was the brightness of the bulb itself, since the current has been reduced by 55%. This is something the OP will have to check out and decide.
Smaller wire won't hurt anything, but it won't have the effect you think it will. This run will probably be less than 5'. As an example, if he's using 5 ft of 20ga wire, and that's a liberal estimate, the wire resistance is only 50mΩ. This won't drop hardly anything. He will still have the same power consumption at the bulb.
That said, I agree using LEDs would be the most efficient method. Evidently I thought it, but didn't post it (I keep reading the blasted thing but it isn't there. Bummer).
Hairyeyeball,
You might consider using LEDs. Since you work with custom bikes, this will have a snazzier appearance. This is another option you could check into unless your dead-set on using the bulb.