Soak a paper towel with adhesive remover, and lay it over the residual adhesive in the METAL parts of the cse, and you can remove much more easily (try razor blade). On plastic trackpad base, use plastic tool.
Also, the torx screw in the center of the trackpad base is your click adjustment. You may want to calibrate that BEFORE putting in the new battery.
I found placing the spunger under the part of the connectors AWAY FROM THE WIRE (top and/or side depending on which of the three you’re finessing), and then rocking them gently from side to side pops them out gently.