This is the type of setup that just needs lumens. You don't need any 4K capability or much else. I would possibly consider the used market for a business class 1920x1200 (WUXGA) projector that is nice and bright with low usage if possible.

$3,000 will buy you nearly 10,000 lumens on the used market, which is a ton, and will do a solid job filling the entire space.

You will need a Chief RPMAU mount, or better, depending on the weight of the projector, and you will need the proper ceiling mount for the room itself to hang it from the ceiling securely and an extension column of the correct length.

Contrast is pretty meaningless when you have a room with that much light in it. But, it will be fine for news, sports, and other 'bright' viewing options.

The list is VERY long for 1920x1200 projectors, and eBay has a ton of them...
Projector Central - Search By Feature

I would also look at Epson's Clearance Center for models that are a good price. I would put an absolute floor of 5,000 lumens in such a room, but you really can't have too much light output in such a bright space. It's really a terrible space for projectors, but with enough light, you can get a workable image. You will need to try difference sized images to see what size is the right balance of brightness when watching your typical programming.

I've limited this Epson search to WUXGA projectors with more than 5,000 lumens.
Epson Clearance Store