I had this same problem as the op. I had a faulty battery. I was very confused by the erratic behavior. I ran a full charging test on the elect. system. Everything checked out at the time. But then on the way home it acted up again. The next day I ordered a new battery.
I pulled the car into the shop to demonstrate the problem to another Tech.He was just as confused as I was and insisted it was something catastrophic. Now pay attention. Here is where it gets weird. As I am loosening the tie down bolts, the car comes alive. I had left the key to the on position (no real reason just did). And as I let the pressure off the battery the short that was being caused by what I found later to be a broken battery cleared, and the elect worked normally again. The plastic case of the battery had failed and the top post was touching the plates inside of it. I must have cranked down too hard when I installed it a year+ ago.
Only signs of anything wrong was the dimming of the headlamps when I hit the brakes and some other weird stuff like that. This was after many months of driving, Even an autocross run...No leaks to be seen and no visual clues to the culprit. This is just what happened to me. You said you put a new battery in along with a whole bunch of stuff. So I dont know...
Good luck I hope you get it figured out.