Just completed this last night. They weren't kidding when they all say the battery is no fun.
I managed to get it, but doesn't mean the old one wasn't bent like crazy when I was done. I was planning on keeping it as a backup since it wasn't bad, but I don't think I would risk it at this rate. Getting a thin stiff plastic to push through the adhesive along with the spudger helped.
There are a lot of components and boards along the edge of the battery. It's hard to pry without touching them. I saw a guide recommend taking the guts out to remove the battery.
One thing I would recommend while you're in there is to replace the thermal paste. Mine wasn't too bad after a year, but I'm sure it won't make a solid and effective cover on the processor to the heat pipe.
Can I just use Artic silver MX4 for thermal paste? Or is there a paste specifically for it?
Arctic Silver should be fine!