The only problem with projectors is there's not much choice if you're sensitive to DLP rainbow effect. I haven't tried one of the newer ones with a faster colour wheel, though. It means I've had to go JVC DLA projectors, but these are now ridiculously expensive and I can't see myself ever spending that much on, well, anything.
Yes, projectors with 3LCD tech is what you are looking for. They produce all 3 colors at once via 3 distinct lcds inside the chassis and mix them ahead of time. There are a few to choose from, but they all cost above 3000.
The reason why projectors don't use a single rgb lcd (like monitors) to produce the color is the same why all sub 5000$ projectors use pixel shift to fake 4k resolution: Too much light is blocked by the lcd itself if the individual pixels become too small.
I am somewhat sensitive to the effect and have been okay with an X3000i. If I scan my eyes across a black screen with white text, I can still perceive the effect - but it's nowhere near as bad as some older DLP projectors.
