hi

the way to reduce colours in photoshop is to change your image mode to indexed color, selecting the number of colours you require; always work on a copy, not the original image

however I would then project that image onto the canvas and draw outlines over it and then colour these in using a printed image for reference

the way you suggest is unlikely to work unless the image is coloured only with flat pure tones, and even then I see no advantage in it

but of couse, without seeing exactly what it is you are working with and what you wish to achieve, I can only speak from my own experience and I have not used projectors for art in a long long while

youtube has tons of videos on the use of projectors in art...