Having resolved the issues with Itchy and Scratchy, we managed to get the Click Wheel iPod dissection done. There aren't any big surprises under the hood of this iPod. It uses the same Portal Player chip, PP5020, found in the iPod mini.
The battery is stuck to the Click Wheel module, which is welded to the face. We didn't bother removing the battery, as it was late in the day and the Click Wheel's ribbon connector, stuck under it, would be too easy to damage.
Update: There is some interesting new stuff under the hood which we missed when first posting the dissection.
Another Update: We went ahead an pulled the battery out. Having removed the logic board, we put the iPod's plastic front in the sun for about 10 minutes. This was enough to loosen the adhesive holding the battery to the Click Wheel module. We've added a picture to the dissection...