This is an interesting question and I am usually asked the reverse at work- retention. Here's the deal, though, cognition and maturity are two very different areas of development. While your son is academically very bright he is likely a typical eight year old in many ways. If he "skips" a grade you are going to put him in a classroom with children 2-3 years older than he, not just one due to the arrangement of the split-level classroom. Essentially you would be asking your son to go from a developmental level of second grade into the socially complex 5th grade without any of the "practice" in between.
You talk about graduating at 16, but that would also mean college at 16. Do you think he would enjoy his freshman year when his mom would need to drive him to class b/c he doesn't live on campus? Look much further down the road... will he be able to date in HS and college? Probably not. Will he have a peer group in HS and college? Probably not. Could this lead to some emotional and social challenges as a young adult- very likely.
Just some things to consider... are you wedded to his private school placement? Could you look at a different private school that could offer him additional experiences and opportunities? Could you look into enrichment programs in the community? I'm not sure where you are located, but there are some great programs at the local community colleges for gifted children. Vassar College (Poughkeepsie) runs an outstanding summer "college" for gifted children as well.
Please consider the social-emotional aspect of your son's life as well as his academic enrichment. Learning to negotiate and communicate with peers is as much a predictor of long-term success as is academic achievement. Your son is very bright, there is no doubt about that, but if he doesn't have a peer group and goes through HS and college without a date and no driver's license, how happy will he be?
Good luck with this one and truly consider a different private school. This model may not be the best for your son.