Had another go with this camera today. No dice: I still couldn't find anything to blame right away.
Out of curiosity,
THIS is the injector it came with. It's the T8124 model. I had a look to see if it gives any info on how power is delivered and apparently it's a gigabit-capable injector, so it uses all 4 data lines - it doesn't send power over the spare pairs.....I THINK....that's why I said I don't quite get how POE works in a scenario like this....