For screen size, the middle of a movie theater will have seats which are 1.5x the screen width away. With a 120" diagonal screen, you have a screen width of 105" - just under 9', which would make a typical 'middle of the theater' seating distance at 13.5'. I would expect at least two rows of seating, similar to above, and the front row likely around your 13.5' 'ideal' center distance, then a second row about 5'-6' behind that. Which would be considerably further back then 'center of theater'.
At this point, you fall in the minority, as many people are viewing at distances closer to 1:1 to 1.2:1 of screen width. A 120" diagonal from 10' viewing distance is more and more common as people like the immersion that it offers. But, as is always the case, this is personal preference. THX standards calls for the 1.5x screen width viewing distance as their standard.
To each their own on this, as long as the rest is good. For front projection, a dark room, with dark carpet, walls, and ceiling and proper light control is a must. Good projectors are a real plus, but good starts for not a lot of cash, and rarely gets better than the JVC RS series of projectors unless you go with the Sony 4K model. Don't get suckered into exotic projectors.
Frankly, if the cost wasn't to much, then I would likely add the extra footage (if the choice is not having it at all), then build a back wall at the proper distance and make the extra space storage. Put your equipment back there, lower the noise floor and light floor in your theater space. There is never a time when an extra 165 square feet of storage space is a bad thing. It's also a great place for that equipment.
Oh, and potentially you could build a false front wall and put speakers behind it with an AT screen.