On the symptoms that you describe, the most likely case is not that you have a battery issue primarily, but an adapter issue. You mentioned that you have tested the socket and think it to be good, but it is still possible that the adapter is not receiving power from the socket, that the adapter itself has failed, or that the connection from the adapter to the console is not conducting power.
It is even possible that somehow the connection inside the console from the adapter connection point is not working.
But as you yourself point out, when things are working as they should it will not complain about the batteries when it is receiving external power.
My experience is with the VP2, I assume the Vue is similar in this respect.