Although the new battery comes with its own circuit board, we'll have to use the old one as an intermediary / extension

Remove the remaining pieces of the old battery connectors from the two main pads. Be careful not to rip off the pads from the circuit board.

To avoid any conflicts between the logic of the old battery's IC and the IC of the new one, we have to remove the old one. Remove the small IC from the circuit board using an exacto knife or similar

Use a soldering iron and some soldering wick to remove the remaining solder from the pads for the IC

Note The pad to the left of the IC (U2) is for the positive side, and the pad to the right of the IC is for the negative side (see last picture for correct orientation)