Just to follow this up, I bought a battery off eBay - under two quid, new and mint condition - and fitted it today. So far it's been charged and started playing several hours ago and still going strong, so first impressions are good.
However... even using the plastic tools supplied with the battery I managed to distort the iPod's metal case with the result that two halves don't quite fit together properly. The job was easy, maybe too easy, which makes me wonder if the iPod's casing was already a bit loose. Oh well, it should last another few months before I have to replace it.