It is absolutely battery related - I had the exact same problem... same hardware, same symptoms, same everything.
I would guess your battery is saying to "replace now" (mine was, though no idea if yours will). Either way, I disconnected the battery from the motherboard inside and everything runs just like new. It is a simple connector to remove; iFixit probably has a picture of it if necessary.
No other solution worked for this -now I am running with "no battery" until a replacement can be ordered.
HTH
Edit: Well, this worked for a short period of time; now back up to near 800% and a useless MBP. Got a battery on order so will see if that does the trick.