Probably not a good idea to remove cables while the engine is running...you can check output of the alternator at the stud on the back of it. 13.5 volts is on the LOW end of acceptable charging, 13.8 - 14.5 is the range I'd like to see. On the higher end if the battery is discharged. With the symptoms described, i'd suspect the regulator, assuming he still has the external set-up. Fully charging the battery and leaving it disconnected for about a week and checking voltage it shouldn't be lower than 12.6 volts...that's a fully charged battery. If it's anywhere near 12.0 volts, it is probably bad...wouldn't be the first new bad battery sold. Anywho, hard to say without the car in front of me...let us know what happens.