You have a kind of scatter shot mess of information here. A car that won't turn over isn't always a low battery, it could just be a bad connection at the battery, or a bad starter, or a bad connection at the starter..
Next time it won't turn over, check your battery voltage directly at the posts. If its anywhere above 12.2, its just a loose connection or bad starter.
When the battery is low, the alternator will put out more voltage, at least the one in my truck does this when i leave it parked for several days.
Then when the battery is well charged, the charging voltage drops to ~14V
So in your case, intermittent charging (or possibly a constant load drawing down the battery when it is parked) puts the battery in a weak state and the alternator pumps out more juice to bring the battery back.
If you have a charger, Charge the battery up well, then re-test the alternator charging voltage.
THe battery reading @ 12.7 is fully charged..
EdIT:
3 month old battery means this has probably been an ongoing issue? or maybe the battery change created the problem?
Either way, often times batteries go dead because the alternator is on its way out.
the old alternator in my truck had deep grooves worn in the contact rings and brushes that were worn too short to make full time contact.
I would notice my headlights going from normal brightness to dim randomly when parked with my headlights pointed at a wall
