I have replaced the battery in my iPod Classic coupe of years ago. I think I followed the ifixit guides: https://www.ifixit.com/Device/iPod_Classic
It is quite a brutal business getting inside an iPod, because the pressed steel back and the aluminium front are very stiff. I used little flat blade screwdrivers, scalpel blades and plectrums to get the blighter open. There are hidden clips inside, and you need to bend the sides of the chrome part out quite far to get at them. I definitely thought I was going to ruin it, but it proved to be very sturdy, and you would never know that I had opened it up. Batteries are cheep on eBay etc, but make sure you get the right one. It's well worth a try if you are a wee bit adventurous.