Trying to decide how to classify a business

You can establish a social/athletic club as a non-profit corporation under state laws. Then you can apply for tax-exempt status with the IRS. You might not qualify as a public charity (donations to your organization being tax deductible for the donor), but nothing you are proposing looks like it would run afoul of tax laws and expose you to unrelated business income taxes from the operations.

I have prepared the tax returns for an organization that promotes and sponsors dance competitions. They qualify as tax-exempt, and I can't see why an argument can't be crafted for your venture to qualify as well.