You've proven that the AGM is not charging the 12v battery. What you don't know is whether the APM is bad or whether the APM is not receiving the power it needs to operate. Do you know whether the logic circuits of the APM use the 12v battery? If so, there could be a fuse feeding it. Or, one of the connections to the APM could be loose or a cable could be broken. Too many possibilities for me to recommend swapping out parts.
When you measured the 13v with the car powered off, you didn't have the EVSE plugged in by any chance did you? That is a bit high for a battery sitting idle.
