Without detailed circuit architecture it is impossible to offer real help. There are just too many variables.
I believe you have a high resistance connection somewhere. Likely where your busses, alternator B lead, master power, master contractor, or etc. are connected. The alternator seems to not be getting full battery voltage when the engine is running. This is why you have the high charging voltage.
The battery is not low. 12.9 volts when not running is OK. A fully charged automotive battery is 12.6 volts and an odyssey PC680 is 12.85 volts.
There is the remote possibility of a ground issue, but would look at other first.
You could power everything up, like the plane is running and hook a battery charger to the alternator B lead at the alternator to simulate the circuit under real conditions. If you can reproduce the problem then check voltage at each major junction. Everything 12 volt should be about the same voltage.
If the meter is connected to the battery negative everything that is ground should have very little to no voltage on it. If there is more than .5 volt (I know a general high number) at any grounds there is a ground problem.
You stated you had the engine running, with the master off, and the alternator was charging. This is not good. It is not a good idea to have the alternator running with the battery off line. I would recommend that the master switch be a dual pole switch that also shuts off the field or control lead to the alternator and to be sure that the alternator does shut down when the field or control circuit is disconnected. If not you may not be able to shut down the electrical system with the engine running.
Also, do you have any high voltage protection. My guess is no. If no, it would be a good idea to install something for high voltage protection.