Well, I hate your room. I mean, it's a solid room. Colors are good. But, that front wall.... !!!!!!!!
I wish it was flipped, you wish it was flipped, we all wish it was flipped, it should be flipped. But, that's a ton of work, and a fair bit of cash, so roll with what you have and enjoy it.
Throw distance is measured from the front of the lens to the screen wall. So, if it's 20' to the back of your room that's about 18.5 feet to the front of the lens (give or take) and if you get a projector like the Epson 5050, then the lens to screen for a 96" diagonal must be between 9'11" - 20'10".
Sounds good right?
WRONG! (of course)
Because it needs to zoom in to fit any 16:9 content you intend to view. That brings it down to a 75" diagonal. (37" of height)
A 75" diagonal can be filled from 7'4"-15'5" lens to screen.
The crossover between those two distances is where the lens MUST be placed. That's 9'11" to 15'5" for a 96" diagonal 2.35 screen.
I wouldn't go any further back than 15' and no closer than about 10'6" with the lens.
All calculations pulled from:
https://www.projectorcentral.com/Epson-Home_Cinema_5050UB-projection-calculator-pro.htm?id_=75
I would really keep an eye out for a JVC RS540 if you can find one. The black color would go much better with your setup.
Be aware, that every projector uses a different lens and has different throw distances. So, you must calculate based upon what you will be buying. Also, not all projectors have extensive lens shift. Epson is, by far, the best in this regard.