If you have the skill to solder a new connector on, the best option would probably be to look for a donor motherboard.

My advice for this step is to remove the black foam pad covering the connector so you can see it better (put the pad back on the connector later). The way I removed the connector was by lifting upwards with the wires.

@beefmangta With a machine of that age assuming the thermal paste has never been replaced, it is absolutely worth replacing. Certain aftermarket pastes such as Arctic MX-4 (which is available on Amazon) also tend to perform better than stock pastes in many cases.

You are correct, this step is not necessary for a battery replacement although it may give you more room to work with. In my opinion it’s not worth the risk of damaging the connector though, as you can just place the metal cover to the side like Rob said, leaving the connector attached and carefully folding the metal cover to the side like opening a book (although it may not lay perfectly parallel to the front case of the iPod).