Richard,
Good timing! The 2023 reservation lottery is just opening up in a couple of hours, is open through September 7 and they let you know in October. Go to the St Andrews website and submit for a 2023 reservation for the day you want. You're required to play a second course, but that would be worth it if you win that lottery.
That being said, September will be tough to do the regular 48 hour ballot. The course is closed September 1-4 and 11-22 for events. Usually that means there's pent up demand for the other dates. However, someone has to win and there are enough other courses in the area that its probably worth trying if the dates align. If you're trying to maximize your chances, Thursdays have a couple extra ballot spots and Saturdays have the most open ballot spots.
Good luck!