Well, I have to say that you seem to have a seriously tough nut to crack here. If the US Open has been played at this club, the club is obviously very exclusive and the members clearly have a lot of $$$$. Clubs like this pride themselves on preventing their members from being "bothered" by people selling things. I can tell you that I golf at a private country club and that my club has an arrangement with a detailer like the one you are looking to establish, but I can't tell you how much business the detailer gets from the club. It's a relatively upscale club and the lot is filled with BMW's, Benz's, Bentleys, Audis, etc.
The idea is solid though. Basically, it takes 4-5 hours to play golf (and maybe eat lunch and shower too) so while the car is sitting there it's a good opportunity to have it detailed. At my club, you call the pro shop who arranges the detail and you leave your keys at the pro shop when you arrive. The detailer picks up & drops off the car and the cost of the detail is applied to your monthly bill from the country club. So, it really is a great service to the members.
I would definitely try to set up a meeting with the GM of the club you mentioned to arrange something like this. I think there are three things you can do to increase your chances. First, present it as a service to the members which will increase their satisfaction with their membership at the club. Second, bribe him!! Offer up a couple of free details either for him or to give away as prizes for a golf tournament or something. Third, offer a kick back to the club. For every detail you do referred to you through them you will pay them xxx amount of $$$. I think the arrangement is great for you, but you will need to motivate them to allow you to do business with their members.