Thats definetly an charging problem (On the rectification side) or a regulator.

With a recon alternator you should get a new regulator as well.

The charge output is fine so long as it can overcome the internal resistance of the battery, the greater the difference the greater the charge. But not to much hence why regulation is required.

As a rule you never want to be seeing anything less than 13V WITH the engine running under load, ideally slightly more with no load.

One last thing that may be worth doing is checking the power from the alternator itself, where the heavy duty read cable connects, just be careful of the pulley/belts with the engine running!

Martin