This was in the mail two days ago:

There were a second sticker and a third board but I just wanted to show top and bottom here.

The board cuts aren't really clean:

The result after soldering (the Aisler stencil did a good job!) and adding a teensy:

In that circuit, I just use the teensy to

• supply power to the power button board
• measure the sense output (pin D0 with an internal pullup)
• measure the output voltage (pin D1)
• control the keep_alive pin (pin D2)

Here's the sketch, which

• prints some state output on Serial
• when the button is pressed, keeps the output alive for seconds

  #define SENSE_PIN 0
  #define POWER_PIN 1
  #define KEEP_ALIVE_PIN 2
  
  void setup() {
    pinMode(SENSE_PIN, INPUT_PULLUP);
    pinMode(POWER_PIN, INPUT_PULLDOWN);
    pinMode(KEEP_ALIVE_PIN, OUTPUT);
    digitalWrite(KEEP_ALIVE_PIN, 0);
  }
  
  void loop() {
    static const uint16_t on_time = 2000;
    static elapsedMillis on_timer;
    static bool on = false;
    static const uint16_t print_time = 250;
    static elapsedMillis print_timer = print_time;
    if(!digitalRead(SENSE_PIN))
    {
      on_timer = 0;
      on = true;
      digitalWrite(KEEP_ALIVE_PIN, 1);
    }
    else if(on && (on_timer == on_time))
    {
      digitalWrite(KEEP_ALIVE_PIN, 0);
      on = false;
    }
    if(print_timer >= print_time)
    {
      Serial.printf("sense: %01d | power: %01d | on: %01d\n", digitalRead(SENSE_PIN), digitalRead(POWER_PIN), on);
      print_timer = 0;
    }
  }

 This worked as expected.

I also tried to modify the circuit to make it latch by itself. It just needed a strong enough pullup resistor (I had 2k7 at hand) from Q2's base to Vbat_out and once the button was pressed, power stayed on by itself.

Next: Test with a Teensy that can shut itself off with this circuit, because that's waht I had in mind when I started this project.