You can't really answer your question without numbers. Any display not intended for stadium use is quite small with respect to the wavelength of the power mains. There will be quite a few dB of attenuation just from the aperture. A screen with a low resistivity layer will provide even more attenuation, but only you can say what good enough is. Do you need 40 dB, 80 dB or 120 dB of attenuation?
It is both easy and hard to find how much you need. Just operate your device without an enclosure in the environment that is pertinent and measure the noise. The hard part is making sure that the noise you observed is field induced since there are many sneaky conductive paths which can affect this.
The pinhole camera idea, or a small film camera if your box is big enough can answer the other end of the question. With the pinhole, or no window at all what is your observed noise. The two measurements will tell you what the window attenuation has to be until you work on other noise sources.