I'm assuming you've been using a charger on battery? A battery with more voltage than alternator is producing defies physics. Their tester can can load test battery & charging system separately, then give a print out. You basically performed same test by checking voltages with with fan running & lights on. Flaky regulator or weak alt can give same results. Unless alt is tested while full fielded, it's a guessing game as to which is causing problem. I've seen a couple alternators with a burned connection to one of the diode banks. They did charge but could produce maybe 15 amps. With more than radio & headlights on, voltage would fall below 12v.
It turned out to be the alternator. I checked it this morning and the alternator is reading 13.45 after installing a new one.
Glad it is getting better for you. 13.45 at idle is OK. Probably want to see closer to that 14 at higher RPM and nothing turned on.
