Not Beta specific.
If it were the belt then the LED should still light the moment the ignition is turned on.
In an ideal world and assuming a 9 diode alternator, the domestic alternator warning lamp would be fed via a relay that is energised by the same terminal n the switch as the engine warning lamp. After the usual hand and eye checks on connections, possibly fuses, and cables, I would suspect the relay.
To add:
Still assuming a 9 diode alternator. If you have the ignition on, engine stationary and the take the D+ (warning lamp) connection off the alternator and touch it to clean metal (negative) the lamp should illuminate. if it does, then the chances are the alternator brushes are worn out. Often they come as part of the regulator.