Thanks Sphig for this awesome discovery! I've been waiting for a solution for this for many years and Google search every 6 months or so to see if anyone has sourced a new battery. Found your thread and ordered mine 12/1, came today and I just soldered it in. Works great!
I only undid one screw (the one right above the scroll wheel). If you take that out and open up the door, the scroll wheel assembly slides off. Then I disconnected the ribbon cable to the scroll wheel and the battery. Pulled the battery out and soldered the new one in (without taking off the back cover). Reassembled and it's good to go.