You can use the nano but you have to know which "pins" to use. This great tutorial is specific for the Uno.

If the old battery was swollen, is it possible it damaged something inside before you did the repair.

The connector from the battery to the motherboard is high-density. If I have time I’ll do a basic voltage probe.

Well, it’s 10 years later, but I just did a teardown of my playbook, and I had the same question. The battery label is really clear, what I didn’t realize is both big black rectangles are batteries. It makes the balance of the playbook very even. They squeezed the motherboard in between. I have it partially apart now, I’m looking to see how the two batteries are wired together, it’s not obvious (at least to me).