OK. So far we know the charging system is not working. What we need to do now is find out why, is it a bad generator not putting out enough of a charge, bad batteries unable to hold a charge, or a bad control module not "telling" the generator to do its thing?
Let's start with the fact that at 12.2 volts (equal to about 60 %), your batteries need a charge. Fully charged those puppies should read 12.6 volts across. Most of all, trying to charge the batteries with the generator might burn it up - it wasn't made to do that.
So, first, we need to charge the batteries. Once that's done we'd be left with either the gen or the ECM as the possible culprits.
The ECM sends a duty-cycle signal to the gen field through the Gray or Tan wire. So, all we have to do is look for that signal to be present at the gen - duty-cycle means it is a digital signal that is switched on and off by the ECM, the bigger the % of duty, the more the gen wil charge.
To check it, you will need either a scope or a voltmeter capable of reading frequency (Hertz), or you can rig a Noid light like the ones used for testing injector pulse. All you have to do is touch the Gray or Tan wire terminal at the gen with the positive (Red) probe of a scope or frequency-capable voltmeter, or one of the jumpers from the rigged Noid light using it as a you would a test light (regular low-impedance test light might not work and instead cause internal ECM damage)
Please let me know what you find and/or if you have any other questions.