Now a replacement battery can be obtained at XINJ store on Aliexpress. It's a 1000mah battery, not 1100mah as the original one, but it would work as a replacement in case Valve refuses to help.
Video with controller disassembly up until the battery compartment can be seen here:
How to replace the battery with a connector (I'd still go for solderinng tbh):
Info about battery can be found here.
PL802056V 3.8V (model number)
1100mah 4.18Wh (specs)
+18100604-1 (this looks to be battery serial number, as I found 2 different ones)
More details in my own disassembly:
10.78x56.86x21.47 in paper fold with rubber pad and expanded chamber and control board (finished battery).
50.59x19.24x9.62 just the normal dimensions without control board and expanded gas chamber (just the cell without wires/board).
Model number under paper is the same: PL802056V 3.8V 1100mAh 4.18Wh
Serial number is different from the paper wrap: 19072902-1