The problem is the intermittent nature but: when you get it to stay on:

Pull the key and then (duh), pull fuses one at a time until the car shuts off.

That should eliminate a bunch of possibilities. There may be more than one device involved (faulty device shared line with another where the fuse pull breaks the connection to hot).

Open ground can easily put weird voltages in weird places so also do some ground wire testing.

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2011 E70 • N55 (me)
2012 E70 • N63 (wife)