What can be done to consistently nudge the sleep cycle so that he
wakes after 6:30 AM?
I don't think there is a reliable way to change his sleep cycle. Sleep cycles are quite personal, and some people (children and adults) just need less than others. There are some things you can do to influence the sleep cycle (later bedtime, lots of physical activity during the day), but this may not always work. It sounds like you have already exhausted this approach, but if not, it's certainly worth trying.
Another approach would be to address the immediate problem - your not getting enough sleep. Some things you could try:
- Discuss the problem with your son. He should be quite old enough to understand - try to find a solution with him. Can he play until 6:30 before waking you? Teach him to read the clock, and talk about what he can do in the morning. Maybe promise a small treat (such as reading a short story to him) if he only wakes you at 6:30.
- Allow him to sleep in bed with you. Opinions vary widely whether and for how long co-sleeping is okay, but as long as it does not bother you, it could be an option. You could tell him that it's okay for him to come to your bed if he is silent and does not kick or push you. However, you must decide if that is okay for you (and your partner, if you share a bed with him/her).
Finally, please try to find an interim solution for your lack of sleep. Lack of sleep causes loads of problems, in particular it makes it hard to solve other problems, because it makes you irritable and reduces creativity.
Try to get a nap during the day, or sleep longer during the weekend, or get someone else to look after your son from time to time - anything so you can sleep. You owe this to you and to your son.