I would be very hard pressed to tell you any specifics of what you would need in terms of setup. Projector brightness is typically based upon the size of the screen, so throwing out a number like 5,000 lumens doesn't mean anything without first knowing the screen size. As well, be aware that unless you are using sourced material at 4K, then having a 4K projector may be useless.

For shooting video, you may also want/need to get a projector which can properly sync up to the video equipment so that scan lines don't appear in the final product. I know all of this is important, but I don't have first hand experience of working with pro-level gear that has these features and functionality.

You also will need to remember that the projector itself will have a fixed aspect ratio. So, if you end up with a really wide background, that's not as tall, you will likely need multiple projectors setup with edge blending.

There is a lot going on here and a lot which could go wrong. I would likely get something to test shots with before committing to a specific projector and a budget.