Use an ATTiny 85 to 'push' momentary button

attiny85.ino

#include <avr/sleep.h>

#define adc_disable() (ADCSRA &= ~(1<<ADEN)) // disable ADC (before power-off)

int buttonPin = 0; // phyisical pin 5
int ledPin = 4;  // physical pin 3

void setup() {
  // https://www.notion.so/spencerowen/AtTiny-85-automated-chicken-coop-heater-596304ab6b3145769b92ddffdeb16b6f
  // https://arduino.stackexchange.com/a/66655/27311
  // https://electronics.stackexchange.com/a/521634/23954

  // set pin to floating
  pinMode(buttonPin, INPUT); 
  pinMode(ledPin, OUTPUT);
  digitalWrite(ledPin, LOW); 
  // wait 2 seconds for electronics to fully power up
  delay(2000);
  // Simulate button press & turn on led
  pinMode(buttonPin, OUTPUT);
  digitalWrite(buttonPin, LOW);
  digitalWrite(ledPin, HIGH); 
  // Hold button press for 3 seconds
  delay(3000);
  // Set pin to floating and turn off LED
  pinMode(buttonPin, INPUT);
  digitalWrite(ledPin, LOW); 

  // Go to sleep
  set_sleep_mode(SLEEP_MODE_PWR_DOWN);
  sleep_enable();
  sleep_cpu();

}

void loop() {

}