AAXA spends a lot of money on advertising, but the reality is that they are like many others... They are making REALLY cheap (but decent) product, and your expectations should match. The brightness of almost every projector out there is over-stated. Much like the contrast. As well, you won't be in a 'theater' space, so your quality will be in line with the space you are using it in. That is, you can't walk into a poorly light controlled room, and shine up a 500 rated lumens (150 real world) projector, onto a wall and expect a great image at 100". But, you may have an acceptable 40" or so image. Acceptable in a lit room, but very good in a dark room. This is the fact of front projection - you can battle ambient light, but you don't beat it, and you battle it with lumens, which these models just don't have. Compare, by example, the BenQ W1070, which is actually capable of delivering a full 2,000 lumens if needed, or can deliver over 1,200 calibrated lumens. That's ten times the brightness that you should expect from these smaller models.
Still, if you don't have the room or ability to work with that larger, and frankly more fragile, projector, then a pocket projector may be a very good choice.
I would recommend the LG PF85U as a first choice as it will deliver some of the most significant lumens out there and is a 1080p model, unlike every other model on the market.
I would shoot for brightness...
There is a rather long list of similar models to that AAXA model from LG, Dell, Asus, InFocus, Viewsonic, and others.
I would start with THIS LIST:
Then see what other reviews are out there on Amazon and other sites.