You really should speak to someone who knows projectors, because you are saying a LOT of things which just aren't realistic and may have a setup which requires you to simply throw out every single thing you think you know and stop saying 'THIS IS WHAT WE NEED AND THIS IS OUR BUDGET', because if you want things to work right, you need to just stop. That is, make a list of your WANTS, your DESIRES, and your BUDGET, then accept that you may not get all of them.
START: Projectors are like TVs. You don't go into Best Buy and say "HEY! I love that TV, but can you give it to me with the same width, but 12" taller?"
Projectors are a fixed aspect ratio. Most often these days, in the commercial world, they will be 16:10 aspect ratio. A very good resolution is 1920x1200 for commercial displays. It is high resolution without being ridiculously high and impossible for people to read. 4K in the commercial market place is stupid, stupid, stupid in almost all cases.
So, if you get a 16:10 projector, and you want a screen 10' wide (DO YOU KNOW THAT IS THE PROPER SIZE EVEN? HOW DO YOU KNOW?), then your 10' wide screen will be about 75" tall. It will NOT be 9' tall. It can NOT be 9' tall. So, throw that out the door. You could search (and find) a 4:3 XGA projector (1025x768 resolution), but that is truly reaching back about 20 years in technology to demand obsolescence and nobody worth their weight will let you buy into this.
Generally, the rule, under normal indoor lighting, is to have about 80 lumens per square foot of screen space for best results with PowerPoint type presentations.
10' wide is a screen of surface area: 62.5 square feet. At 80 lumens a square foot: You are right at 5,000 lumens. If you've done the actual math on a 4,000 lumen projector not being enough, then I expect that you may be happy with any projector that can deliver a true 5,000 lumens or higher, and more than happy at 6,000 lumens.
I would go with LCD, period. They deliver the color brightness you need and are more consistent in their delivery of a bright image.
Now, the fun part: You have a long throw distance, demand wireless, and a ridiculous budget.
Just search! Make a requirement for your search 6,000 lumens, 1920x1200 resolution, LCD projection, and that it.
The LEAST expensive model is running $3,500.
Now, because you demand that the projector is 37 feet from the screen, you will not only need a projector with interchangeable lenses, but you will need a long throw lens for that projector. That's going to be $1,000 all by itself. That's reality. So, there's what you would consider half your budget.
Let's look for a value and get you there... Check the Epson Refurb website and you get something like this...
With this model, you would need the Epson Middle Long Zoom lens, ELPLM10.
Which, in the real world, all in, is about $3,500.
Oh, and wireless projection is crap. You will need power at the projector location, and you should plan to wire a PC and provide a laptop connection or two to the projector using cat-5e extenders. They are a few hundred dollars each.
You will need a decent screen for this as well, if you don't have it already.
Then an installer than knows what they are doing.
I don't see this all happening for a lot less than $10,000. But, for the projector itself, with your long throw distance, you are stuck at a minimal price of $3,500 because of your throw distance.
If you could mount the projector within the range of the 'standard' lens, then the refurbished projector ships with it at your target price.
So, something needs to give. Your throw distance is the one that gives you the capability to hit your budget. Or, move the budget, and you can hit your throw distance.