Battery seems to be working fine, there are no issues with starting, when the car is not faulty, and both mechanics I went to with the problem, as well as the AA recovery guy, said that the battery is fine and there is no evident problem with that. I've had a few attempts of resetting, disconnecting and reconnecting various things, with no result (even found a method for resetting BSI module, but no success). While I aim for a miracle to happen, with little idea what I'm doing, nothing helped so far.