Are you sure that the low battery warning isn’t referring to your fobs battery? I’ve never seen a low battery warning that referred to the main battery, only the fob battery.
You may want to check out the owners manual...
^^^ This. I don't think I have seen a "low battery" message on the bike in any other context except the active FOB or the tire pressure sensors
[update- there is, indeed such a message, see followup posting, below]. To the OP: When last did you replace the battery in the FOB? They typically only last about 2 years.
Of course, based on symptoms reported, he could be having issues with both the FOB and the starter battery. Pay no attention to the "Battery" voltage with the engine running- that tells you nothing about the battery, but the voltage of the alternator, mediated by the battery. And it doesn't tell you as much as you might think when not started, either. You can't really tell the overall condition of the battery based on voltage at rest, it must have a significant load on it. The best test is removing the battery and placing it on a large tester that will charge it completely, let it rest several hours, then throw a huge load on it and measure the voltage and amerage at that load. That said, if you were getting 12.0V BEFORE starting the bike, on a battery that is supposedly fully charged, then you certainly have a bad battery. 12.0 = less than a 25% charge on a lead-acid battery.
The Concours is very picky/particular about battery condition and battery connections and frame grounds being clean. Based on many years of watching this forum, I would guess that 90% of all electrical problems and strange behavior are based on those factors. I have experienced it, myself.