Random thought: the Christie projector might be very loud (fan noise). You might need to build a hush box for it or locate it in a different room (wall cut out with glass window light pass through).
With a $6k projector budget, I’d have advised an Epson LS12000, Sony XW5000ES, or a JVC NZ7. That gives you a 4K system with good to great native contrast and superb dynamic contrast, and good HDR tone mapping. Paired with a normal screen, you can have 120” to 160” image. And all with reasonable...
200” is a huge screen. Even with a something like a JVC NZ900 you will be on the dim side for the image. And you need around 20 feet of throw distance for the projector.
So if your heart is set on a screen in that realm, you’ll probably need to spend some time playing with projector calculators...