One idea is a 64 tooth ratchet wheel which is driven from the pendulum pushing system.  This would be connected to the strike shaft through 6:1 gearing and a "free wheel" (like on a bicycle).  A lever/pawl from the pendulum drive would be arranged to engage with the 64t ratchet wheel during the wind-up period - being started by a cam on the hour shaft and stopped by the weight being wound to the top of its travel.

Oops - just realised the strike wheel and shaft rotates anti-clockwise when striking so the winding has to be clockwise.

Might change the timing to the stepper motor to give a rotation speed of 2s per revolution and save a gear or two.  I can do this with an Arduino sketch - here is a first attempt.

// Filename :- Pendulum_Longcase_Clock_v2_2019-11-14
// Software timing from RTC using polling
//
#include <DS3232RTC.h>    //http://github.com/JChristensen/DS3232RTC
#include <Time.h>         //http://www.arduino.cc/playground/Code/Time  
#include <Wire.h>         //http://arduino.cc/en/Reference/Wire (included with Arduino IDE)
//
boolean lastSqWave = 0; 
boolean ledON = 0; 
int sqwPin = A6;
int stepPin = 6; // STEP pin
int enPin = 12;  // Enable pin
int ledPin = 13;  // Internal LED pin
//
void setup() {
  pinMode(stepPin, OUTPUT);
  pinMode(enPin, OUTPUT);
  pinMode(ledPin, OUTPUT);
  digitalWrite(enPin, 0);   //  enable
  pinMode(sqwPin,INPUT_PULLUP);   //  RTC timing pin
  RTC.squareWave(SQWAVE_1_HZ);    // 1Hz square wave            
}
//
void runClock(void){
  for (int i = 800; i >= 1; i--) { //  count 800 pulses in half second ie. 3200 micro-steps per two seconds
    digitalWrite(stepPin, 1);
    delayMicroseconds(10);        // Make STEP pulse 10μs long
    digitalWrite(stepPin, 0); 
    delayMicroseconds(600); }     //  1600×610 = 976ms
  ledON = !ledON; digitalWrite(ledPin, ledON); }  //  flash LED on & off (for testing)

//
void loop(){
  boolean sqUp = ((analogRead(sqwPin) > 500));  // read logic level of 1Hz square wave
  if (sqUp != lastSqWave)  { lastSqWave = sqUp; runClock(); }  //  Call runClock on both edges of RTC square wave
}
// End

To allow for the different timing the motor gear can have 16 teeth instead of 25 and drive the pendulum and strike auto-wind as well as the main clock.

Also, following from my Giant Wall Clock where the gears from the stepper motor that drives the seconds wheel produce a lot of noise from the teeth, it occurs to me that the motor gear and 64t epicyclic drive gear will probably also make a noise.  This is pointing back to a ratchet system where any noise would be on a 2s cycle like the clock itself.

Going back to basics :-

1. The chain wheel rotates at 4x the centre wheel which rotates once an hour.  So chain wheel takes 15m per revolution.
2. The chain wheel (sprocket) has 18 teeth and the auto-winding sprocket has 8 teeth.  So AW sprocket rotates at 15x8/18 m/rev = 15x60x8/18 = 400 s/rev
3. Drive motor rotates at 2 s/rev so overall ratio required is 200:1

This could be achieved with a 40 tooth ratchet wheel and a 5:1 pair of spur gears.  This seems a lot simpler than the epicyclic gear system.  (OK so I've changed my mind again - my prerogative! 👩🤣 )

New assembly so far.