It could still be the battery. If you are charging high amps the voltage will be low, hence the 12.3 volts reading. It's the old flowing water analogy. A shallow river flows fast (voltage) whereas a deep river (charging amps) flows slower.

Load test the battery. If it tests okay then re-check your alternator. If you have an induction ammeter you can check for running amps. You should be showing 13.5 V and about 2 to 20 amps at idle with no accessories on after the battery has recharged the starting amps draw.