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.
Happy to help! I think it comes down to budget too. If you have $1000 to spend, you can get a 4K projector that can work even with dim lighting. But if you only have $300-400, then you’re starting to look at slight tradeoffs - picture quality might be worse with a projector, but then the screen would be bigger and it would be easier to move around.
If you know anyone with projectors, you should go try one out and get a feel for the differences!