So… turns out my APA102 strips aren’t so bad after all. After some discussion with the wonderful Phil Howard on Pimoroni’s forums, he confirmed that my approach should work, and either my joints were dodgy (I didn’t think so, I’ve done quite a bit of soldering along the way), or the data and clock wires were the wrong way round.
Just to be sure, I redid the soldering, alternating the wires on either side of the strip, like so:
the theory being there’d be less chance of shorting between the pads this way. Still no joy. So then I tried switching the data and clock wires (the photo above actually shows after they had been switched). And oh joy, it works! Here’s a strip of the APA102 LEDs working together with 3 Mote sticks, running off the Mote controller board: