Hi Tim,
I don't have a definitive answer for you, but I'm thinking you might be able to accomplish what you're trying to do.
The battery actually consists of two parts; the lithium ion cell itself - the part that holds the charge - and a small circuit called the BMS, or Battery Management System. What I'm thinking is if you separate the BMS from the cell and solder that back onto the iPod, it may allow it to boot.
It seems the iPod really wants to talk to the BMS in order to boot, so having one installed just might satisfy that requirement. Of course, the iPod won't see any voltage coming from the battery, so there's always the possibility it'll shut itself down due to a low battery, but if you have one of your old dead batteries around, you don't have much to lose by trying it.
The picture below is from an iPhone XS, but the principle is exactly the same; locate the plus and minus contacts on the battery cell and cut them off, leaving just the BMS to connect to the motherboard.
[image|2861093]
If nothing else, it will certainly be an interesting experiment.
If you give it a try, be sure and come back and let us know what you find.