Ok; I'm a little confused , but that's normal at my age .
You're getting alternator voltage " everywhere " with the engine running .
Inverter panel shows 13.8 volts , 5 mins after engine shut down .
When do you get 10.2 volts at the fuse panel ???
What is the voltage AT the battery when the fuse panel has 10.2 ?
13.8 volts is a surface charge at the battery , if after an hour or so you're only seeing 10.2 volts ; I would suspect a dead cell in the battery rather than a bad connection.