IMO, as you drive the vehicle the alternator warms up and causes the idiot light to come on and you say with a gauge it is only putting out 12 volts when hot and that indicates to me that there is either a cold solder joint in the regulator or on one of the diode legs or one of the diodes is weak,
There are other possibilities as Roger pointed out, the alternator could be too small for the load and there could be a short in the system, though that should show up hot or cold.