if you want it to do ambilight just send a second video output to your projector, set monitors to duplicate screen, line up the center of the projector throw on your screen and blur with the focus ring on the projector, all it's doing is increasing the size of the screen output and the blur takes away any of the definition so it's just colours.
a project ariana would be much more difficult,
https://livestream.com/studio
Vimeo Studio software for live effects, OBS to live capture your screen into it
from screenshots and videos Ariana takes the live footage (OBS in this case) adds a distortion effect to match the screen dimensions and then extreme stretch the rest like an ultrawide lens distort effect (probably going to be calibrating the projector throw manually and digitally to line everything up with an X and square so lines don't jump too badly onto wall) and project it onto the wall/screen and wall depending on where that projector ends up being placed. (in your case behind you for the extra throw distance)