I re-soldered the RP2040-Zero board right-side-up, but the programmer still wasn't working. After a shameful 2 hours of debugging, I finally realized I must have fried some magic pixie gems inside the RP2040 by soldering it and running it the wrong way around, so I popped a new one in and it worked a treat. Project complete! See video of it working. If I make another one of these, I will try to do the toner transfer better. It might even warrant an investigation into the heat and pressure levels that give the best results.