You're in a very bad situation really. The screen is a 4:3 screen which is outdated, your throw distance is exceptionally short, and your room situation stinks.
I would ditch any attempts to fill the screen completely and just go with a current day widescreen projector that is as bright as possible and works as best as possible.
So, 16' wide by about 9' tall and you have about 150 square feet to fill. That's large. That's FAR larger than typical and is common for about a $30,000+ projector. Very common.
At a minimum projector rating for a lit environment of 80 lumens per square foot, you are talking about a projector with no less than 12,000 lumens - or about 4 times what you need for a typical classroom environment.
If you really want the 4:3 aspect ration (16'x12' screen) then that's 192 square feet, which calls for about 15,000 lumens.
Here's the list of projectors over 10,000 lumens with interchangeable lenses to help hit your VERY short throw distance.
Just to keep things a bit closer to home, I've kept pricing under $50,000.http://www.projectorcentral.com/projectors.cfm?g=1&hide=0&st=1&mfg=&p=1&p=50000&w=&r=&br=10000&br=50000&ll=<g=&t=&db=&dt=&c=&ar=&dvi=13&wr=&ol=1&pjl=&pjw=&pjh=&td=&i=d&is=&sort=pop&sz=15
Basically, half the projectors on this list are Panasonics, and for good reason.
The real 'budget' find in this list which I would probably go with is this one:http://www.projectorcentral.com/Panasonic-PT-EX16KU.htm
Not sure of your setup, but with their shortest throw lens, you can get your 240" diagonal from 12'10" lens to screen, which is very close. Not sure if the lens has offset though and would work properly. Other options are to shoot the image from across the room, or to use a lift or other feature to get what you need done.
Be aware, that projectors have lamps which must be replaced and have a cost associated with that and must be accessible to change out those lamps, so I would expect a projector lift would be necessary.
Anyway, not sure of your budget, but very realistically, under $20,000 is something of a bargain for the materials.
There are better solutions out there, but I would expect the projector to be between $60K and $100K.
All projectors work fine when hooked to a PC or BD player as long as they have HDMI inputs, but you will need to address the audio as well as the projectors won't provide the audio feed for the space.
- Theater, whole house audio, and technology consultation during the build and installation process in the Washington, DC metropolitan area.