I see you obviously took my counter to your recommendations personally and you countered with some ridicule and silly arguments.
I am not going by MFR lumens, considering I wrote a projector calculator and had to collect real-world lumens of many projectors, I would say I am as familiar of the REAL lumen output of projectors as anyone (and more than most).
I am also fully aware of how a projector overshoots non-native aspects, and that is exactly why I would not do it when starting from sub-720p resolution with a 4:3 aspect onto a 16:9 screen (you will further reduce your resolution). There is no reason to buy a 4:3 format projector for home theater viewing in a backyard unless he just wants to exclusively watch 4:3 movies. 2.35 movies will be terrible from a 4:3 aspect projector that starts at 1024x768 and then projecting it with the huge loss of resolution caused by the overshoot of 4:3 to the inner 2.35 conversion on the 16:9 screen (rather than 16:9 at 1080p or 720p, to 2.35). You also lose some SERIOUS lumens when using the over-shoot method for such far off aspect ratios, so none of your lumens arguments even hold up. I am fully aware of marketing contrast lies with on/off contrast, which is why I currently own a JVC RS-45 LCOS projector which can truly approach 50k:1.
Depending on what he wants to spend and his exact lighting conditions, the Panny ar100u is the best choice to maintain an HT look to the image with some reviewers measuring just under 2500 lumens in its brightest mode (and 2000+ lumens in "accurate enough" modes), or there are several Epson LCD's in 720p, or he could buy a 16:9 or 16:10 business projector. He could also paint a positive gain gray screen that rejects ambient light, that will do more than just getting a brighter projector.
Arguing about optics when citing the resolution of 1024x768 in LCD with all the crazy conversions you are recommending doesn't hold up to say the least, not to mention the much worse pixel fill ratios from the older panels at 1024x768 res, and then projecting non-native aspect ratios and losing pixels from aspect overshooting and adding ringing from scaling, then adding the conversion of 4:3 to 2.35 (I am speechless). The average VISIBLE convergence of optics is partly based on the pixel size and resolution (as far as how the human eye can see convergence error).
Then you attempt to make a case for reliability issues when comparing a used projector to a brand new or refurb that comes with a warranty when all problems are fully covered. Then you attempt to say that 4:3 is better because of old movies (many old movies have been converted to wider aspects, and the OP will most likely be watching wide aspect the majority of the time). I'm sure the neighbor is going to enjoy his IMAGE overshot into their yard given this poster was planning on using a 16:9 screen... This person is not trying to install a projector for a media event at a football game or Charlie Chaplin movies from a seating distance of 200 feet back, he is installing it to presumably watch primarily 16:9 and 2.35/2.40 movies in his backyard near his pool on a relatively small screen.
If he just wants the absolute cheapest setup, then it would be hard to tell him what to buy since it would vary based on what was in the refurb or used market on any given day. To be sure of any recommendation I would have to see how much light is hitting the screen, but the ar100u with a positive gain gray is a good bet, and it may even work just fine on his current screen with no changes depending on his lighting. I did not see the OP mention any desire to watch mostly old movies.