You could try “inverse…with respect to x” and see if that improves matters.

This is easy

Solve[ y ==  Exp[- Exp[-(a-x)/b]], b ]

Attempts to use InverseFunction on 'alpha' did not work for me.

InverseFunction applied to expressions with more than one symbol does not work, of course. The function has to be declared as a map and for the symbols regarded as the coordinates of the manifold to be mapped. Position in the sequence of variables and its total lenght have to be given in order to constract the partial inverses as a function. Its a similar problem in the algebra of functions of several variables in Derivative, if switching from the mathematical pictorial display to algorithms.

      InverseFunction[Function[{x, u, g}, Exp[-Exp[(x - u)/g]]], 2, 3]

     Function[{x, u, g}, x + g Log[1/Log[1/u]]]