So, you are fighting at odds with reality.
"I want something bright, small, silent, high resolution, and super cheap... with plenty of zoom and lens shift."
YES! That's what we ALL want.
It's completely unreal to ask for, and doesn't exist. If you find something that makes the claims for what you want, when you visit their website you will find out that they don't even have a manual for you on their site, let alone a support phone number or a good email address.
The best, cheapest, 1080p, nearly silent projector on the market comes from XGimi. It's not exact tiny, and it is nowhere near $200. More like $900...
Amazon.com: XGIMI H2 1080P Full HD Smart Projector 1350 ANSI lumens 3D Home Video Theater Projector Support 2K/4K with Android System WiFi Bluetooth Beamer Harman/Kardon Speaker: Electronics
www.amazon.com
Plus, it STILL doesn't have lens shift or zoom on it, so you get zero placement flexibility.
If you want placement flexibility, zoom range, a bright image, low noise, and 1080p resolution, then you should get the Epson 3100 or 3700 models.
Still - $640! WHY? Because that is simply the price of a high quality 1080p projector with the lens capabilities you are asking for.
Bring home blockbuster performance with the Home Cinema 3100 projector. Offering optimized brightness and contrast, this projector features 2,600 lumens of color brightness and 2,600 lumens of white brightness1 - plus up to a 60,000:1 dynamic contrast ratio. It's an ideal choice for your living...
epson.com
Kinda hitting more of a budget point, the Epson 2150, used, via eBay may be had for under $300. It should be reasonably quiet, plenty bright, has some lens shift, and some zoom range. Doesn't touch the 3100 in terms of ... everything. But, with 1080p resolution, it's a solid model and can be had for a reasonable price if you look around.
Find many great new & used options and get the best deals for Epson Home Cinema 2150 1080p Wireless 3LCD Projector at the best online prices at eBay! Free shipping for many products!
www.ebay.com