Here’s my guide on getting it open: Casper Glow Light Battery Replacement I was having trouble finding the correct replacement battery and haven’t finished repairing it myself, but it would involve soldering and may be a little tricky to reassemble
If you try it, please post the results. I talked to one of the people who worked on it and he said that you could use "any lithium ion battery that can work in the same voltage range, but there are a few gotchas like how to safely attach the connector". I don't know too much more than that and I just haven't had the time to learn out to attach a battery like this.