I've completed the circuit and the coding, and it works perfectly. It detects when the countdown has reached one second, then "presses" the Stop button twice, which cancels the countdown and returns the microwave to idle. Those presses produce two short beeps as an indicator that cooking is completed, but they are not obnoxious.
The protoboard is shown in the attached picture. It includes an 8MHz Arduino Pro Mini, a TLC374 quad comparator acting as a level shifter, and an N-channel mosfet which shorts the row and column of the Stop button. 5V and ground are available on an ISP header on the microwave's controller board. Inputs include the four digits' common anode pins. Four of the segment lines are also inputs. Since I only need to detect a blank digit, a zero and a one, only four segments are needed to identify those states to the exclusion of all other decimal values. They are the A, B, D, and G segments.
The common anode lines are a bit of a puzzle because they swing through the full 5V, and because they are active low. I suspect these inputs are actually the gate drives of P-channel mosfets powering the common anodes.
The oven powers the A, B, C and D segments one digit at a time, then goes back and drives the E, F and G segments in the same way. So to get the currently displayed value requires eight readings over a period of 10ms. I'm still not clear why the designer did it this way, except possibly to reduce peak current draw since no more than four segments would ever be powered at the same time.
For the code I was going to use interrupts, but since the Pro Mini has nothing else to do, I decided to just process the pin values directly. My code requires two successive readings of the one-second state over 20ms before it recognizes that state as valid.
This is a pretty elegant solution for me, and it took a bit of work to get right. Particularly helpful were pictures taken at various shutter speeds revealing the two-stage display algorithm. Anyway, it works like a charm. So that's one minor daily annoyance removed from my life.
Netflix showed no interest in doing a documentary on this project. Oh well.
Edit: Details here:
https://github.com/gbhug5a/Microwave_Anti-Beep_Mod