Despite the error in the LCD wiring, I did get the backpack controller working. I put a hack in the library to rewire the interface in software to match the error, but once it's up and running, it's good to go.
The button is the 'start' button for the oven.
It functions identically to the older controller design. While it's running, the temperature on the left is the current actual temperature, and on the right is the current target temperature just below the time within the current reflow session (generally the set point drifts gradually across a phase from one point to another, with the different phases effectively representing different line slopes).