The 'search by feature' function of the main website is something you should familiarize yourself with.
You didn't specify whether you were using a 16:9, 16:10, or 4:3 screen, but you definitely have some issues with the longer throw to that size screen.
Under $3,000 and here is a list of the 17 projectors which can do this size, sorted by brightness.
It is a mix of all aspect ratios, so you should choose the aspect ratio which matches your situation. There are some widescreen options which will do it for close to your budget.
This model from Panasonic looks very promising for your needs if you are going widescreen...
The 4:3 options are almost more difficult because you will need to purchase the lens which will add a bit to the price, but may be feasible near your budget.