On the other side the battery is soldered to the board.

It is marked with T (temperature), B-, and B+.

The battery is a Synergy AHB110520CPS. An online search revealed nothing. I wonder if this is a proprietary battery only sold to Bose, or its just already obsolete (1.5 yrs old).

The battery specs are 495mAh 3.7V, 4.2V max (I think).

It is an Advanced Hybrid Battery, which is a very compact battery technology made by Synergy Taiwan.

Replace with a battery of similar specifications, and try to find one with similar internal resistance. I am not sure how to measure that for this battery but you may look it up online.