Let's start by examining your screen to determine the actual size you are looking at.
It appears that you have a 144" wide 2.35 screen.
If you aren't using an anamorphic lens, then any projector you purchase for home theater will be in the 16:9 aspect ratio, and to fill it edge to edge will have significant overspray of unused light/image above and below your screen.
So, you will actually have a 144" x 81" image size you will have, regardless of what height your screen is if you aren't using an anamorphic lens.
That works out to a 165" diagonal, which is FAR larger than the typical home theater screen (92" to 133" diagonal).
But, for lumens measured in square footage, the math is pretty simple: 144" times 81" is 11664 square inches, which is 81 square feet of screen space to fill.
I would shoot for 15 real world lumens per square foot as a minimum requirement, which means you need no less than 1,215 color calibrated lumens on screen for acceptable results.
COLOR CALIBRATED LUMENS!
Color calibrated lumens have really been brought up a lot lately because DLP doesn't play by the same rules that LCD and LCoS must play by with this. DLP can introduce a clear segment into their color wheel which artificially boosts brightness at the cost of color. This makes the projector brighter for white fields, but not for colors, and actually cuts the amount of color light output which is possible. Typically by as much as 66%. The Optoma HD25LV, by example, claiming 3,000 lumens, actually only delivers around 1,100 or so color calibrated lumens. Nearly a 66% drop in brightness when watching video material.
This is why reviews matter and getting good measurements in reviews matters and why color wheel speeds and segments for DLP matter a great deal. The IN82 you have measured out to a maximum light output of about 1,300 lumens in its brightest usable mode. When calibrated down to a normal screen size, it only had about 680 real lumens. That's remarkably close to the claimed 1,500 lumens it reports. It likely is using a 6x color wheel based upon the price and when it was released.
Sony is using a 265 watt lamp, and likely isn't getting a lot more light from it than what you already have, but in testing it delivered over 1,300 lumens. About the same as the IN82. So, you should expect very similar results, from a new lamp and with the projector on full power. Realistically, neither are suited for the size screen you have except when the lamp is new, and when it's on full power. But, through the use of a low gain screen (1.3-1.5 gain) you should actually be able to establish good lumen numbers for standard viewing at the size you are working with.
PROJECTOR CENTRAL'S CALCULATOR: The PC calculator is a tool that isn't often used properly. It's a reference, but isn't a gold standard for image size or brightness. I've caught several calculator issues for size as well. What you need to do is go back to the reviews, the real math. Look at what the measurements were for a projector, for any projector, and decide if it is enough for the screen size you are using.
I use the W1070 which delivers over 1,200 color correct lumens. I use it with a 1.3 gain 161" screen, and this is what it looks like in a room that is less ideal than yours is (see attached).
The Sony projectors are fairly bright, but they are meant for 16:9 image sizes of about 92" to 133". When you push the size, you want to add some gain. You want to be sure to mount the projector towards the front of the available zoom range it has to increase light output as much as possible. You also want to understand that the Projector Central calculator from 8 years ago was not setup the same way it is today. Today's calculator does not work from the manufacturer claims and derates the output from a projector by a certain amount.
Do the math if you question it.
Baseline: 96"x51" screen - 34 square feet - neutral gain screen (1.0)
PROJECTOR CENTRAL - SONY - FRONT OF ZOOM RANGE
The claim is 25 lumens per square foot.
25 times 34 = 850 lumens from the projector.
Projector Central derated the claimed lumens by 50%, even though the review says it can deliver over 50% more light output than that.
PROJECTOR CENTRAL - InFocus - IN82 - FRONT OF ZOOM RANGE (36 square foot screen @ 96" width)
The claim is 42 lumens per square foot.
42 times 36 = 1512 lumens from the projector.
Projector Central did not derate the projector at all. It is using the actual numbers supplied by the manufacturer instead of a middle of the road, middle of the class, measured rating from the projector.
The IN82 calculations are MUCH less honest than what the Sony calculations are.
I'm not sure there are any 4K projectors on the market which can fill a screen the size of what you have without going to a much more expensive class of projector and do it really well for the lift of the lamp. But, with a positive gain screen and a regiment of replacing the lamps every 1,000 hours or so, you may get results you can live with.
This is all a game of balance for projector manufacturers and if you fall outside the bounds of 'standard', then you will pay for that. This is the cost associated with such a large screen. But, you are getting hooked on the lie of the calculator, which you shouldn't let yourself do. The Sony can be brighter than the calculator shows, and the InFocus is a lot less bright than claimed.