Getting songs from iPod classic with low battery
It depends on wht you mean by "iPod Classic." Which is why we recommend people include model numbers in their questions. That said there is an iPod called the iPod Classic which was the last click wheel iPod Apple sold.

Yes you might be able to get the songs off if it, depends on how much effort and $$ you are willing to expend.

The iPod Classic (A1238) came in a variety of HD sizes but they are all about the same for getting into. EG difficult and finicky. Once open it is possible to replace the battery. It may even be possible to extract the HD and plug it into some kind of external (USB) case if you can find one with the same connector as the HD. I think it is a non-standard connector. Other folks here will no doubt correct me on this...

Lots of places online have instructions on how to do this. YouTube is one, there are others. My favorite is iFixit.com that has instructions for replacing both the battery and HD.

It does not look simple but if you are adept at this sort of thing it looks like it is quite possible to do.