Am I likely to get permission at a Private Airport?
Can't hurt to ask!
I don't know the specifics of the situation, but I would be willing to bet that your chances are better than you think.  If it was one farmer with a grass strip and no accommodations for visiting aircraft you might be out of luck, but from what you describe it doesn't sound like one extra airplane on the ground for a few hours is going to make that much of a difference.
In fact, it might be that Fred (or whoever) will also be visiting friends for Labor Day and might be willing to let you use his ramp.
They can't say yes if you don't ask.