This is full daylight, a short throw projector with 4k lumen """""might""""" be ok, but there is no contrast whatsoever. Looking at the picture, I wouldn't be confortable doing anything projection related there.
You might be better putting up 4 55" LED tvs and arraying them in a 2x2 cluster. You will have enough space, brightness, depth, budget. The only issue is the non-seamless look of the flat screens (in your 4k budget, higher budget and you could get a 100" near-seamless viewing surface).
How big is the 'large screen' and does it block out the sun or is it turned facing it?