You're looking for something like the old red/blue 3D glasses effect? That is, you see the blue image through the red lens and the red image through the blue lens, but when you look at the page without the glasses then everything is sort of blurry? You can get the same separation of images with well-chosen red and blue LED lighting as you describe here instead of filters. Plus, you can also add black or gray for things that don't change at all between images to increase the impact. The classic fluorescing "invisible" ink shifts an illumination wavelength from UV to somewhere in the visible range. There are various UV inks with best spectral response at different wavelengths, but there is a tendency to overlap.
The basic idea there is to use two process inks with well-separated reflectance and control either the illumination or filter the visible image to only get one image or the other. This can be done with something like a child's "spy" light to see UV inks only where illuminated or a "magnifying glass" with the appropriate filter.

Another option might be lenticular sheets where the underlying images are arranged in alternating stripes of a frequency appropriate to the lenticular sheet used and a lenticular overlay. You would see a different image depending on your viewing angle. However, fine details tend to get lost and you need a solid printing process to keep the images aligned under the sheet.