Actually a hygrometer isn't a bad idea, to check the state of charge of each of the 6 cells in the battery. I have seen bad batteries mis-diagnosed as alternator problems. I have also seen bad batteries mis diagnosed as a battery that was not charged, when in fact it had a bad cell and would not take a charge.
As mentioned, you should have in the mid 12V range(12.8 is typical for a battery with a reasonable charge) on the battery setting there before startup. If it is below 12V then that may be an indication of a battery going bad(cells not charging), or one that is not being charged because of a bad alternator.
When running, and charging properly, the battery voltage should be at 14.5VDC +/- 0.5 VDC. If it is low, it could be one of 3 things.
1. The alternator is bad and cannot produce the rated output voltage.
2. The regulator is bad, and cannot tell the alternator to produce the rated voltage.
3. The battery is bad, such as with a shorted cell, and is absorbing more energy than the alternator can output.
It is a system, and these 3 parts interact with each other. One way to eliminate a battery issue is to charge the battery using an external plug-in charger, or swap it out with a known good battery from another vehicle.
If you have a known good charged battery in there, and you are not getting your 14.5V at higher RPM, then you have an alt or reg problem.
Your manual saying it only makes 5V at lower RPM, leads me to believe that you have a generator and not an alternator. Alternators usually are capable of making rated voltage, even at lower RPM, but they cannot put out as much current at lower RPM.
Good Luck