just use a voltage regulator to take the voltage back up to 12V
The voltage regulator you are thinking of only reduces the voltage to 12v. It will not boost it to a higher output than the input.
I would tend to think that there is some problem with the wiring in the car, and "making do" with what you have isn't going to solve the root problem. Maybe the switch is bad? If the contacts are dirty then you might see a lower voltage than normal.