To starts with the engine running you should see voltage at about 14 volts not 12.6
I would venture to guess you have a bad alternator from the parts store. Not to uncommon to get a bad alternator out of the box. Crappy rebuilds. Anyways start there.
LOL I didn't read your last sentence, sorry. That is a good question, I believe that you would have issues driving that vehicle without the IP cluster in it. I know on other vehicles I have working on it does cause issue. Some vehicles will not shift properly, some will not drive at all. So yes I would wait to get the cluster back. I would imagine having a BCM in the vehicle would cause issues if it did not see the cluster. When you get the cluster back recheck you output again and see if it goes up to the 14 volts it needs to run properly.....