With that kind of budget and a dedicated room you want a JVC. If you can find one a RS540 old stock would do very well but a new full 4K NX7 is around your budget. Treat the room with dark colors especially the ceiling and with an inexpensive screen like a silver ticket it will come as close as you can get to a OLED in front projection. Always buy the projector first and use the wall or temporary screen for a few weeks to find out what screen size/AR/seating distance works for you and the room before even considering a screen. Projectors in dedicated rooms blow any kind of TV out of the water. In less then ideal rooms is where it gets very complicated trying to use Front projection.
"Smart enough to know better, to old to care" ------ Dedicated Bat Cave Home Theater, JVC RS49U/Mitsubishi HC7900DW Projector, 110" 16:9 Jamestown screen with variable power masking for CIW 2.50:1 to 16:9, Marantz 7009 with 7.1.4 Atmos with Ohm mains,3 DIY Subs (2 15" (1 ported, 1 sealed and a 12" 4th order bandpass), 1 DIY butt kicker, Custom Built HTPC, 18TB DroboFS NAS