I just had a similar problem. Mearured the battery with the car off got 12v, with car running got 12.4v. Shot electric motor cleaner in alternator, replaced regulator, cleaned all connections. Now I have almost 14v at battery with car running. No more dead batteries at night. Paid $18 for the regulator at Autozone. Be sure to remove the metal shipping pin looped through the brush wires
It might not be a bad idea to pull the alternator and have it tested.