Well, there are options here. You could assume that the time value for the wealthy travelers is high and the time value for the poor travelers is low. Your objective is to maximize the (revenue - cost_to_maintain) while retaining some max_visitors. You could then offer the Faroe Islands option: you get to trade off some degree of your time and some degree of your wealth to visit. So you could either pay $1000 to go to Preikestolen or you could help maintain the trail for some x days etc.

Ultimately, let's be honest: we want the most number of people to be able to experience these things (I fortunately went to most of these places before the recent boom and I think they're wonderful) while ensuring that the cost to the environment isn't catastrophic. That means some people can't go. We can pick some mechanism but it will necessarily exclude some people. The money is the best selection mechanism because we can use it at pretty high granularity on preserving the thing that is bearing the cost, but any substitute that serves the purpose does just as well.

The key magic is that with money we can undo some damage. That's what makes it a useful thing to auction on.

How is this different than what for profit airlines, hotels already do within the tourism industry?

Less wealthy travelers go at odd times, they live in hostels and have "worse" experiences.

The only difference in this case is that the country is also directly profiting from this demand.

I thought that was the point given their goals. I guess it makes their inner ugliness too blatant though.

Vickrey auction?