Can Compton scattering work with any other forms types of photons, and why was the the Compton Scattering experiment only done with x ray photons?
The physics works fine on free electrons at all photon energies.
Now ask yourself
The loophole I've alluded to is to use an electron beam, but people don't use it because it is rather more trouble than just using ionizing photons in the first place.
Compton Scattering takes place not only with x ray photons but also with gamma photons. You may test this with a gamma emitting radioisotope like 131-I, 57-Co or 137-Cs and a scintillation detector commonly available in any nuclear physics laboratory. Keep the sealed source with low radioactivity in one hand but close to the detector. The source should not be in direct view of the detector. Keep the other hand few inches above the detector. You would notice a rise in count rate by the scintillation detector due to detection of Compton scattered gamma radiation from hand. Using this principle, I have developed a technique imaging human body by Compton scattered gamma rays in 1974.
There is a reason why Compton scattering work with X-rays and gamma rays and not at energies lower than that of X-rays. The X-ray or gamma ray would be scattered when passes through core Coulomb space. However, the energy of X-ray or gamma ray should be sufficiently high for Compton scattering to take place by core electron. For example, when 300 KeV energy gamma ray is incident on human body, it gets scattered at an angle from incident beam with less energy. Since sufficient energy is required for Compton scattering it does not takes place, say with other types of photons, say light photons.