I’m dealing with this now. I leave an old second gen in the car because it has a 30 pin adaptor in it. The temp drop got it.
I have it charging now and it appears to ''slowly'' be charging. The frequency of the Apple screen appearing has decreased and I can actually read the “No power to battery remains. Please connect iPod to power.” I didn’t even get beyond the logo when I first plugged it in.
I’m going to let it charge for a bit and see how it does. My only concern is that the older ones like mine have a tiny hard drive and every cycle, I’m hearing it activate.
My advice thus far is to just be patient and let it try to charge. I’ll post an update if I can figure out how to stop the restart cycles.