If it is private, it is not a true religious enterprise, but rather a private one, meaning that Congress may pass laws prohibiting activities or the exercise of those private activities and any and all tax breaks received by said mosque and any "clergy" should be rescinded! They can't have it both ways.
I'm sorry, but that is just ridiculous. You must not be thinking through what you're saying. Catholic Churches aren't private? Of course they are. They are tax-exempt because they promise not to engage in political dialogue, that's all.
But, the Church controls who is or is not a member, right? Have you ever heard of excommunication? Do you think you have right to appeal your excommunication to the court? Of course you don't. Why? Because the Church, like all churches, is a wholly private enterprise.