Immersive room design plus other small questions

Hello everyone. My brother saw some videos on tik tok about creating an immersive room such as having a snowy background in a room wall. Now he wants something like this.

What is a good projector I can get him for this? Does it need to be short throw for a room, I believe it’s maximum 10-12ft? How come some Amazon projectors say ANSI 800 plus lumens, 4k for ~$300 but then real name brands I’ve seen are like 200 ansi lumens with 1080p for $3000? Preferred something on $300-600 range. Needs to be able to cast to it for wireless display. Thanks!