Have to disagree. A battery that is putting out 12 volts is undercharged. Even showing 12.5 volts is low especially on our bikes. He should be getting close to 14 volts at 3000 rpms almost instantly, not 30 seconds at 5000. I would have the battery checked first, then check the rectifier.That's fine.
Is the battery the "AGM" type?I should mention the battery puts out around 12.5 with the bike off. The battery is several years old now. I've heard it's not uncommon for our bikes to discharge at idle though.