Barco.... ha, ha, ha... Seriously? No, please... tell me you are joking. Next you'll suggest Runco...
People do outdoor setups on a budget all the time and with minimal ambient light, from twilight on, a decently rated projector which can deliver the light output will do just fine on a screen. It's not a guessing game, it's math. 16'x9' is about 150 square feet. Looking for 13-18 lumens per square foot in a theater environment at home, which means it only needs a bit over 2,000 lumens to be acceptable in a 'dark' situation. So, a recommendation of 5,000 lumens is certainly not a bad way to go while still delivering 1080p video and doing it for under $2,000 which is appropriate for the section of the forums we are in.
Would a Panasonic projector spitting out 8,000+ lumens be even better? Absolutely! Likewise, we can get a Digital Projection, or a Projection Design model as well, but then we are a bit outside the reality of the budget implied by posting here, and from the original post, it doesn't sound like it is at all appropriate.
If light is generally controlled, other than moon light, then it is reasonable to expect about 25 lumens per square foot of screen space is a realistically usable amount of light to produce a very good on screen image (not a poor one), and that would be about a 200 square foot screen, which is larger than what is being proposed in the follow up.
That said, I agree completely that you do want to get the brightest projector you can for the money, make darn well sure it is a 16:9 projector and preferably 1080p, and then blow it up to the size you want to achieve and see how it looks. That will let you know how big your screen can really be while still delivering an acceptable image. I would think 16'x9' is well within that range under most normal outdoor settings.