High energy photons scatter off electrons via Compton scattering. The differential cross section of this scattering is given by the Klein-Nishina formula, which contains within it the classical radius of the electron, $r_{e}$,
$$\frac{d\sigma}{d\Omega} = \frac{1}{2}r_{e}^{2}\left(\frac{\lambda}{\lambda’}\right)^{2} \left[\frac{\lambda}{\lambda’} + \frac{\lambda’}{\lambda}-\sin^{2} \theta\right]$$
Have the equivalent experiments been done with the proton? Is there a "classical proton radius"? The closest equivalent I can find is the proton charge radius, which stems from electron-proton scattering.
Edit: The differential cross section of high energy photons hitting protons found here (Figs. 10 and 11) looks quite similar to data from Compton scattering off an electron (pdf, Fig 2). This seems to suggest both are governed by similar equations, meaning there is a classical radius for the proton.