Heya, yea I have done this myself with my picture frames.
I started with taking apart the battery itself, and wiring to the contacts. Sometimes this would work, sometimes not
I tried wiring 3.7v directly in. This would boot as far as booting the kindle framework then reset (monitoring over serial).
I figured the current draw was what was doing it and added capacitors. It was flakey. Sometimes it worked, sometimes it didn't and it was too bulky to fit inside the case.
After a loooooad of testing, it seems the easiest way is to just hook 5v usb directly on to the contacts. Please see the first image which is with capacitors and diodes to drop the voltage down from 5v.
The second and third rows are the setup I went with. It could do with a top on the pin channel things to hold them down, but it works well.
Is 5v going to damage it? I don't know. Possibly. It hasn't broken any of the ones that I am playing with so far. YMMV. Don't sue me if your house burns down (tho tbh I figure it's safer than dodgy 10 year old lithiums)
Happy to share the models for the latest clip and for the old setups (there were many) if of interest.
Oh, one thing to note, you first need to boot it with a battery to be able to add the battery fail override
/etc/init.d/battcheck
# Don't check battery if we're on prototype/reference hardware
_SKIP_CHECK=1
Kind regards