Thanks a lot for that, Jeff. I believe I actually used a switching 5v supply, which turned out to be enough.
Would it be a safe-ish strategy to aim for a regulated/switching power supply that hits the lower end of the battery pack's voltage? I had imagined it would just draw as much current as it needed as long as I kept the voltage under control, but I hadn't considered the battery being a sort of active power regulation component in the way you're describing. Super interesting, there's always more to know.