Mentch, charge the battery to 100%, then check the battery voltage with the bike running.
Fully charged battery bike off 13.56v. Bike running at idle 12.79v.
What is the voltage at 4000 RPM, starting with a fully charged battery?
13.5v fully charged at 4000rpm
If the battery isn't charging, how did its voltage get higher when you revved the alternator?
The spinning revs have an output relationship.
Not sure about the 650 but other SOHC4 alternators make about 1/3 of their max output at idle.
That's why the voltage sags at idle because the alternator can't overcome bike system load at idle rpm. When the bike revs it can make more than the bike uses and the extra is used to recharge the battery.
Your battery is depleting at idle because the bike is either using too much power or the alternator is too weak at idle rpm to make bike load power and charge the battery.
What has been altered on the bike to use more power than the stock one used?
Higher watt headlight perhaps?
If your particular 650 spends more time at or near idle with full electrical load, then yes, your battery will eventually deplete to exhaustion. That is what your reported numbers show, and what is fairly typical of the entire SOHC4 fleet. If you have selected a battery that has too little storage capacity, it will deplete quicker than a stock one when the alternator isn't being revved as expected.
Watch the voltage at the battery change from idle and increase the rpm slowly to 4000. At some point the battery volts will begin to rise off that low number. That will tell at what rpm the alternator makes enough power to begin battery restoration. If you must operate at too low an rpm, get a bigger battery. You aren't going to change the alternator design characteristics with a replacement stator.
Cheers,