Alternator or VR issue?

So I have been trying to chase an issue with my battery not showing it is charging. When I test at the battery with the key off I am at 12.6, running around 12.3/12.4 never appears to be charging. Below is everything I have done and just looking for suggestions.

Is this a bad alternator even though tested ok at store?

Is there another test I should be doing to verify what is truly the problem?

Thanks Don