There are a few projectors with 16:9 panels, which would be a perfect choice for your screen shape, but none (
AFAIK) have the sort of light power required to illuminate your screen within your budget. Those with enough power tend to use 4:3 LCD/DLP panels. This is OK for DVD and W/S based TV images, but PC and other 4:3 shaped images will be too tall for your 16:9 screen height. The result will be image spilling off the top and bottom of the screen.
I have considered a couple of solutions. The most elegant is to find a projector with a zoom lens capable of filling the width of the screen for 16:9, and then zooming down to shrink 4:3 images so they fill the screen height. The downside to this solution is the time it takes to zoom and refocus, but on the upside this solution preserves the maximum resolution for PC presentations and the image becomes brighter as the size decreases. This is exactly what you want if presentations are made with the stage lights on.
Philips has a projector that would work well here. The ProScreen PXG20 with optional long throw lens. This is 3000 ANSI lumens, XGA resolution with power zoom and focus. It would come in well under your budget. Other manufacturers may also be able to offer a solution.
The alternative is an electronic aspect ratio controller. This would compress 4:3 images from PC and video to fit the height of a 16:9 screen. The benefit is that there is no zooming or refocusing, but the PC resolution suffers and the image gets no brighter when compressed.
I have discounted using an anamorphic lens option. IMO the optical distortion would be too noticeable on your screen size.
You should also budget for some sort of video scaler or deinterlacer. An iScan would be the budget solution, but a Quadscan or similar scaler would be better because it has multiple aspect ratios and RS232 control. A Quadscan wouldn’t resize PC images though, so don’t use it if you go for the alternative solution.
To anyone about to suggest HTPC, please remember this is a commercial installation where ease of use (by some non-technical staff) and reliability are key. I know HTPC can be made to do wonderful things, but in this application it may introduce more problems than it solves
Regards