The solution is the same one used by airlines and hotels: dynamic demand-driven pricing, charging enough that you don't sell all your seats until the last minute.
An airline that sold out of all its seats 6 months ahead of time would be considered incompetent, but governments do this all the time with reservations for various activities, because they are trying to "be nice" by charging low entry fees.
Increasing prices will increase equity. By charging rich people more, you can afford to give greater subsidies for marginalized groups and poor people, and lower the price from whatever it is currently for them. You can even use that revenue to pay them to go camping, if you really want to!
In any event, that unlikely to be helpful in this specific situation, as the majority of all revenues from rec.gov go directly to BAH[1], rather than to the public land management agencies responsible for stewarding the resources in the public interest.
BAH also has price setting authority, with a dearth of public oversight.
It's called "public land" because it is already paid for, and effectively owned, by the public. I would consider a government who resorted to market solutions "incompetent".
An airline that sold out of all its seats 6 months ahead of time would be considered incompetent, but governments do this all the time with reservations for various activities, because they are trying to "be nice" by charging low entry fees.