The recommended AX200U is the correct projector for flexibility and to deal with some ambient light. At under $1,500 it meets your budget, and you are SERIOUSLY mistaken on the image quality you will get from a projector like this.
These photos:
http://www.avintegrated.com/lighting.html
are worth taking a look at as they represent the output of a 4+ year old 540p projector on a 106" screen.
For mounting, you could mount the AX200 at the back of the room - but at 22' throw distance, you must have at least a 110" diagonal screen.
No doubt though, at a 15' viewing distance, you may want to consider about a 10' diagonal screen. It's not unrealistic, and you will have (suprisingly) excellent results.
My biggest issue with this is your screen. Not sure what you expect in a 'fold up' screen. I would more likely ditch the TV if possible and go with a fixed mounted screen on the wall to preserve the acoustics and layout of the room - using a Carada 120" diagonal Brilliant White screen.
Roll-up screens are alright - but you really will find that cheap roll up screens start getting waves in the fabric very quickly... forcing a need to replace it after about a year. A tab-tensioned, motorized screen is a great option, but will cost a fair bit more (likely more than the projector). The big perk is that they work incredibly and they last forever.
Not sure where you are located so you can see some of these things first hand, but there are definitely options available to you at all sorts of price points, yet the AX200 will offer a certain level of flexibility that you really can't get from any other projector. Really, the best projectors on the market may deliver 10% more quality overall... For thousands of dollars more.