What you say is probably true and your planned fix could very well take care of your problem.
One additional test that you might try is to read the battery voltage at the battery with the probes actually touching the battery post to make sure the battery voltage is dropping when you hit the starter and not dropping across battery cable connections either at the battery or at the other end of the cables.
The reason I suspect this possibility is because of your reading of 12.7 volts in your initial testing. I wouldn't have suspected that with a bad battery. If the battery voltage does not drop, look for a bad cable connection.
