Fixed it myself
My Kindle started working erraticly. Freezing intermittently, not starting up cleanly, just not working properly. Although it appeared to be charging, and the problems were not to my mind obviously power-source related, it did not seem to be holding a charge for as long as it should.
Amazon was unhelpful. After talking me through a factory reset, they offered to sell me a "reconditioned" kindle, for rather more than I was interested in paying, plus shipping. I don't trust that "reconditioning" involves any more than swapping out the battery, and who knows what abuse an unknown unit has suffered. What they couldn't sell me, oddly enough, was a kindle battery.
Never mind, I can go to the local cellphone-and-tablet repair place and get them to replace the thing. They don't have the battery in stock, but say they can order one and install it. Two weeks! Sweet!
Three months and several iterations of "Sorry we can't find a battery anywhere" and I just took the unit back unfixed. I resigned myself to using my tablet as a reader for a while, although I much prefer the e-ink to a backlit screen for simple text.
Wondering just how difficult it could be to source a Kindle battery, I googled "kindle" "battery" "replacement", and clicked the first link that appeared - a .co.nz URL, despite all their prices being in euros. They claim to have stock, but I suspect they're a dodgy drop-shipper making claims they can't back up.
I google them, expecting to see ScamScamScamDoNotTrust warnings out the wazoo. I am surprised when I don't.
With a little trepidation, I ordered a battery and put in my credit card details. I figured I could always file a dispute via Visa when the battery doesn't turn up. Again, I am surprised, this time when a box containing a battery turns up less than a week later. It's even the right size and shape.
Following some directions found on Youtube I levered the back off the kindle - that was easier than I suspected it would be, although having good strong fingernails helped - took out the old battery and screwed the new one in place. Snapped the back back on and gave it a good long charge. Kindle's going fine, now.
Makes me wonder what the cellphone shop's issue was.