First, you aren't getting 'native' 4K for $3,000. It really starts at $5,000 for Sony and JVC native models. But, there are many pixel shifters out there from LCD (Epson) and DLP (lots).
The Epson 5050UB is the hands down best, under $3,000 projector for a dedicated space. It fits the budget... well, in the USA, and from 13' can throw a 133" diagonal, which is pretty much on point. That's RIGHT AT THE EDGE of the throw range, so you may want some flexibility in your setup if that's the case. The Epson handles movies, 3D, and gaming very well with low lag time, great brightness, and superior black levels.
Find screen size and throw distance for the Epson Home Cinema 5050UB projector.
www.projectorcentral.com
On more of a budget, the BenQ 3550 is a popular choice. It works very well from your throw distance and is a solid choice for the entry level home theater enthusiast who wants to experience 4K. Which is a nice way to say that it is ENTRY level and won't match up to the black level performance, or brightness, which the Epson delivers. It also has more limited placement flexibility with the lack of significant lens shift. So, KNOW YOUR OFFSET!!! (look up lens offset if you don't know what it means). The 3550 must be placed about 3" above the top edge of the screen if ceiling mounted.
Find screen size and throw distance for the BenQ HT3550 projector.
www.projectorcentral.com
Other options include the Epson 3200/3800 models which are reasonable and compare nicely to the BenQ 3550 as entry level 4K models. They include fairly significant lens shift which may be needed. They can also hit from the throw distance...
Find screen size and throw distance for the Epson Home Cinema 3200 projector.
www.projectorcentral.com
Finally, if you can find it online, and if it can ship to you, the Epson 5040UB is the best deal going on a budget. It has some limitations on the connectivity with 10GB/s HDMI instead of 18GB/s HDMI, but it accepts 4K/30Hz signals from a source, and works with HDR signals like you may find on UltraHD Blu-ray disc player. The black level performance is very similar to the 5050UB, but the price is the real selling point.
Check their website, they show up at about $1,200 online...
Epson Home Theater Projectors have up to 3x Brighter Colors than leading 1-chip DLP projectors
epson.com