If there are fairy lights, it might affect the color a little bit, depending on the brightness of the projector. This depends on your budget - anything decent should be fine, you just can’t get the cheap $250 Chinese ones. It’s also a solveable problem, since you can adjust the brightness/direction of the fairy lights.

Having to move and store it would be the biggest concern for me. A projector is easier to pack and unpack, but has the issue of a separate projector screen.

If you’re able to, say, use one flat wall as a makeshift screen, I’d personally choose the projector.