VPL-VW665ES

It might be worth checking out. I went from a 2400 lumen Epson to a 1500 lumen Sony and it was a step up in terms of image quality. I have remained a fan of Sony's projectors since. Sony's lens optics and picture processing are better than what's found on Epson and more of the lumens are usable. The 665 can display HDR content and you'll be in even better shape if you're using a Panasonic 420 or 820 as a player, their onboard HDR processing is great.

I work in film post and highly recommend Sony’s SXRD projectors. They are native 4k, with 3-chips - so no color wheel or rainbows. Also, Sony’s lens optics are the sharpest I have seen. Sony’s can be calibrated to perfection but have very good color accuracy out of the box. The lamp-based VPL-VW665 and VPL-VW715 can be found used for your price range. The laser-based XW5000 is also a good option, they are newer but some are showing up open box etc.

I went the open box route, got a 4K Sony and never looked back. Sony optics and picture processing are the best available and their projectors are true 3-chip systems and native 4k.

Open box JVC or Sony will be your best option at that price. They are both true native 4K with 3-chip light engines - so no rainbow effect or pixel shifting. Personally I prefer Sony for their optics and image processing. I actually kept my first Sony 4K in storage after upgrading. It was a demo unit which I got for under $3K.