First of all, the Mist Monastery itself is independent of the Frost Crystal Kingdom, and with its immense power, it doesn't need any recognition.
Not to mention, they've been there longer than the history of the Frost Crystal Kingdom—rather, it's the Frost Crystal Royal Family that are the newcomers.
Secondly, the Misty Mountains occupy a third of the entire Frost Crystal territory, and it even has a natural harbor, but it's a barren land ravaged by Demons, full of crises.
The Kingdom had thought of developing that area, but after three or four rounds of attempts over a period of more than fifty years, they only managed to push the residential areas forward by a few dozen kilometers, and it's still far from the main range of the Misty Mountains.
Since it's completely an undeveloped wilderness area with no actual control, giving it away wouldn't have any impact.
Why not seize the opportunity to do a good turn?