I have the identical issue. I have a ipod classic 5 modded with a 256gb SSD and had a 2000mah battery. It never charged. Replaced the battery at a local tech shop with the standard battery. Still not charging. I have it rigged to a 5000mah battery pack which fits in my hoodie pocket so its portable. This has the wolfson DAC and sounds better than my phone and other DAPs.
I image whatever charges the battery, that is regulates the charge needs to be replaced.