The 12v system does not go through the AVR. It most likely is the Battery Charging Rectifier has failed. This controls the voltage from the 12v feed from the alternator and charges the battery.
1) Follow the wiring from the 12v socket/connectors until you come to a Block rectifier. It will be held on by a bolt in the center and have 4 wires connected to it. If you look at the pins you will see one is seated at a different angle to the others. Test the feed to this pin, you should get a result of between 12 Volts DC to 16 Volt DC.
If you are not getting these results then the Block rectifier has blown and will need replacing. http://www.generatorguru.com/rectifier-98406168/
2) If all is testing fine above, then you will need to test the Battery Charging Rectifier as you have an Electric Start and Battery on your unit. This is a metal block located on the back of the engine, next to the starter motor and it has fins running across it. ( to see an example of this part look up your generator on this page http://www.generatorguru.com/all-power/
) Again this should be giving you between 12 Volts DC to 16 Volt DC.
3) Next,check the voltage going into the Battery Charging Rectifier from the alternator. You should get between 12 Volts DC to 16 Volt DC.
For replacement parts please see our web page. You will need the full model number to do this.: