Only a suggestion since I'm not trying to upgrade my player.

I'm wondering if some aluminum foil would be useful here. Stick the stack together with a bit of double side. Then take two strips of foil and slip them between colums of pins, fold back and around. Admittedly this may be easier said than done. The alum, if it will stay in place, should keep the solder from bridging. Hummm, some more two side tape, not near the pins, could help. Maybe all that is needed is a piece of foil with a slit in it. Tuck the slit down around the column of pins...

I hope some of this helps.