I've tested an early prototype of the equipment to fix a resistor in place on a test circuit. While I haven't managed to, the voltage readings indicated that copper did indeed grow inside the channel, just too slowly for now (I only used a 5V gradient). I also identified other issues:

• Bath resistance increases over time on narrow channels like this, probably due to anion depletion at the cathode. Periodically replacing the bath there with solution at the anode restores conductivity.
• Fluid volume at the cathode is too low, a larger reservoir would act as a
  buffer to provide more anions and reduce purge frequency.
• Copper at the anode eventually chips off, and the small pieces of copper can get into the channel.
• Capillary forces are important at this scale, must take into account.

Right now I'm building a better tool, that offers 230 V pulsed electroplating (yes, just some freshly-rectified AC straight out of the wall), smarter monitoring and more control over bath flow, as well as a modular design that can be simply plugged into the piece.