Many of the most "exclusive" clubs in the USA regularly have outings, essentially renting the course to outsiders for the day.
These clubs have membership policies that do not discriminate against particular classes of citizens and, therefore, they do not fear the implications of public access.
"limited non-members were permitted to use the club on Mondays, when the course was closed to members....in addition the club also permitted the golf teams of several high schools and to use the course, during limited, nonprime hours."
"..Club members testified that, on occasion, they brought guests to the club and that their businesses sometimes reimbursed them for expenses involved in such occasions...All members testified they joined the club for social and recreational attributes, and not for business purposes, and emphasized that they viewed the club as a refuge from, rather than an adjuct to, the business world".
"The Club regularily permits some nonmembers to use its facilities, for a fee...the club also obtains income, on a regular basis, from fees charged for the use of its facilities by nonmember invited guests."
"Because of the involvement of defendant's operations in the variety of business transactions with non-members discussed above, the club properly must be considered a business establishment within the meaning of Section 51"
Justice George, writing for the Court in the case of Warfield vs Penninsula Golf and Country Club, 10 Cal 4th 594
That sets the line in the state of California which you can not cross, by law, and remain a private club-- Hence the law is rather clear in this state--If you wish to remain a private club you can not, for a fee or not, invite outside play.