First, may I start off by saying that I understand how difficult this must be for you intellectually and emotionally. You know your child needs CPAP, your doctors know he needs CPAP, but your son doesn't really understand the vital importance of it, and no one has a solution to the problem of how to get the child to accept the mask for the whole night.* I'm so sorry.
Who is helping you manage your child's PAP needs? Some research shows that a team approach provides the best results for successful PAP therapy. I don't know what country you're in, but in institutions with a high standard of care, e.g the Sleep Center at The Children’s Hospital of Philadelphia, the team includes pediatric sleep technologists, behavioral sleep specialists, social workers, psychologists, nurses and board-certified physicians in sleep medicine. If you've not met with any of these professionals, you might try to seek out their expertise.
You state that setting a prize next to his bed that he gets if he wears it all night has been successful one night at a time. If it works, it works! Some have suggested a sticker chart (child picks and applies the sticker) is also helpful, particularly if there's a substantial reward for (x) number of stickers. It need not be something material; it could be a family outing, a trip to a nearby amusement park, a fishing trip, a sleep over, a trip to the ice cream parlor, whatever your son cherishes, but it should be more immediate than remote (fewer stickers required to cash in for the reward.)
Another possibility is to keep your child busier and awake (fewer/shorter/no naps) during the day so that when he does go to sleep, he is more likely to fall into a deeper sleep more quickly and less likely to remove his mask. (You'll find more like this here).
It might help you to know that you're far from alone. There are support groups for parents in your situation. Whoever seems to know the most about the issue, whether it's your doctor(s), the sleep lab (or whoever evaluates the fitting and function of the mask), or other involved persons, ask them for information about support groups that might be helpful. Not only can you get emotional support there, but parents who have struggled/are struggling with the same issue may have some helpful suggestions.
Many machines can remotely alert someone when the user's O2 saturation falls or the mask comes off. While it will not increase your son's acceptance of the device, an alarm notifying you of a problem may help you to insure he gets more hours of PAP in per night (more is better). He may also wear it more if he's asleep when it's reapplied (I don't know; you're in a better position to answer that.)
Patient education is helpful in compliance with CPAP**. Though your child has Down Syndrome, has he been offered the opportunity to really learn from others (not just Mom and Dad) why he needs CPAP? Normally, one-on-one clinic visits, follow-up telephone calls, telehealth interactions, and group meetings are helpful. Perhaps if he meets with others in his age group who are trying to educate themselves, it will help.
Unfortunately, I have very limited experience with trying to increase compliance with CPAP in children with Down Syndrome. I wish I could do more, and I hope someone with more experience sees this question. There isn't much literature (though there is some) in this area. If you want to search the literature, I suggest you use Google Scholar rather that Google, using phrases such as "improve CPAP Adherence in children". You'll get more reliable information that way.
*Most people - even adults - are not very compliant with PAP (CPAP or BPAP). In one small six-month study with 29 children, one third dropped out of the study, and the remaining subjects had suboptimal compliance.
**Education alone, though, is responsible of only about 30 additional minutes of compliance in adults.
Novel Aspects of CPAP Treatment and Interventions to Improve CPAP Adherence
Improving Positive Airway Pressure Adherence in Children
Correlates of Pediatric CPAP Adherence includes Trisomy 21 patients.