I think this is an interesting question. Unfortunately, many hasty sketches of the history of physics, as they are taught, tend to draw somewhat biased conclusions for the sole purpose of avoiding delving into these types of questions (some people consider it to be a waste of time apparently).
As far as I can tell, the classical scattering theory at the time was essentially Thomson scattering theory which is basically an elastic scattering limit of the Compton scattering.
Since it is based on the assumption of an elastic scattering, it cannot explain what is observed for Compton scattering where, if anything, the frequency of the light is observed to change.
Now, to enter into a proper inelastic model one must try to figure out where goes the energy of the wave and propose an actual model for it, especially when using classical wave theory; and that's the difficult part.
In looking for references recently, I came across this paper by Raman (the same gentleman as for the Raman effect) who concludes his paper in a retrospectively surprising way by saying:
Incidentally, it becomes evident why the argument of the "triangle of
momenta" by which Compton obtained his formulae gives the same result
for the change of wavelength as the classical theory of the emission of
spherical secondary waves by the electron.
Although there is no doubt that QED does a great job describing light-matter interaction better than any theory has done so far, it is thus safer to refrain ourselves from setting sharp demarcation lines with historical experiments allegedly signing the death warrant of classical theories; it is always much more complicated than we think.