Thank you all for sharing these tips; yes, this also resolved my problem: the wire connecting the negative pole of the battery was torn, i.e. as it was attached to the spring on the battery compartment door and yes, it's a DESIGN FAILURE to have this tiny and very weak wire soldered to the end-point on the door. Of course, it broke as per the 'paper-clip repeated bending' analogy shared above already.
I don't have soldering equipment at home, so had to improvise to fix as follows:
1/ Very gently peel the wire-casing to expose the metal wire inside, I had to remove the door to more easily reach this tiny wire
2/ Make a 'cap' with aluminium foil to cover the battery's negative pole: this cap should reach half-way through the battery, such that when it's in place it can make contact with the exposed wire
3/ Re-attach the door and close it, to ensure the exposed wire touches the aluminium cap, you should keep inserting extra bits of aluminium folds until the closed-door presses the wire against the aluminium cap that covers the batter.
It brought my headset back to life! It works. Just need aluminium foil. I will complain to Bose about this, too, very predictable failure point.