Hi @vellah ,
Don't know your headphones but perhaps the wire is a "sense' wire to tell the control board to switch the battery over from supplying the headphones to being charged instead or something similar.
As there is a different cable to connect the transmitter base to a computer than to connect to the headphones for charging, it may be like when you have to use an OTG cable in Android. This "tells" a device like a phone (or tablet) to supply power to the port, e.g. when inserting a USB flashdrive, rather than to receive power from the port.e.g. when connecting to a PC. This is done by supplying a permanent -ve signal (eth or ground) on a sense wire connected to the port. Long way of saying it tells the control board to act differently.
If you can, with the "charge cable" plugged into the breakout pcb but not connected to anything, use an Ohmmeter to test if there is a connection between the "ID" connection and ground (USB -ve) on the breakout pcb
If there is then it means that when you insert the cable it puts a -ve (or earth - ground) signal on the ID or "sense" lead. In your case you would think that you'd have to connect the yellow wire to the "ID" connection in the breakout pcb.
If you connect the yellow wire to the ID connection you could also check what voltage, if any, there is between pins Vcc and Ground on the breakout pcb when you connect a s/c between ID and ground (or plug in the micro USB end only of the charge cable if it has a s/c between pins 4 & 5) and when you don't have anything plugged into the port or a s/c between pins 4 & 5
As I said I don't know your headphones but can they still be used if you are charging the battery? If not perhaps it is a signal to turn off the bluetooth to allow the battery to charge
This is all conjecture on my part so apologies if it is totally wrong.