The Battery light is an indicator for the altenator only, it does not come on if the battery is stuffed.
The light excites the alt, and if it is not regulating the reg must have given up the ghost.. If you remove it it will be cheaper for you, just tell the auto elec that she is charging at 16volts and let him go from there. Nothing is worse then someone bringing in a alt/starter and saying "this dont work" without acctually seeing what its doing.
Saying that the reg is stuffed might also be incorrect as most alts have a voltage sensing wire, and it has blown a fuse or has a break in the wire, the alt will detect no battery voltage and therefore charge flat out.
Cheers
Sumo