It works! So leaded or unleaded solder is currently set by changing a variable to 'leaded' or 'unleaded', then re-uploading the software. Next step is making the pushbutton set the temp profile. There's no PID involved, this program just repeatedly checks the temp and turns the coil on or off, which seems adequate for this application. I've never seen it overshoot by more than 2 degrees C.
K-type thermocouple is held to the toaster by Kapton tape, which has shown no difficulty at the temperatures involved (max 218C for lead-free solder).
Engineering notes are here: