Problem using iPad mirroring with an Apple TV and 4:3 Projector

I'm a teacher and I've been using my iPad 2 in the classroom for about a year and a half. I would use Apple's VGA Adapter to connect to my projector, which worked fine. I just didn't like being tied down, and the dock connector would come loose very easily and lose connection to the projector, which interrupted the flow of the lesson.

So this summer, I got an Apple TV and a projector with an HDMI port so I could start using Airplay Mirroring to get the iPad screen onto the projector. But I'm having trouble with the aspect ratio. Forgive my drawing, but the genius at the Apple store had a hard time understanding my explanation, so I'm hoping this will help show what I mean.

My projector (ViewSonic DLP PJD5133) can be set to either a 4:3 or 16:9 aspect ratio (4:3, 800x600 is the native resolution). My projector screen in the front of the classroom is 4:3, so when the projector is set to 4:3, the image fills up the screen (which is what I want - it needs to be as big as possible so the entire class can see it clearly; plus it means all the output pixels are being used). When the projector is set to 16:9, the image is letterboxed (not ideal unless I'm showing a widescreen video or image).

The Apple TV's native aspect ratio is widescreen, and the iPad's native aspect ratio is 4:3, so if you use Airplay mirroring on a widescreen TV, the Apple TV will pillarbox the image (add horizontal black bars on each side), so that the image will not be stretched/distorted. (I know that it will show widescreen videos and a few widescreen-ready apps without the pillarbox, but the apps I use in class - Keynote, Noteshelf, etc. - don't fall in that category). The effect on my projector screen is a tiny 4:3 image taking up half the area it would have taken if I had used the VGA connection.

So in the Apple TV, I went to Settings>Audio & Video>TV resolution, and chose 800x600 60Hz (the max 4:3 resolution for my projector). Since the Apple TV is natively widescreen, it seems to accomplish this change be stretching the image vertically; but it basically works because the Apple TV home screen fills up the projector screen. However, when I turn on Airplay Mirroring on my iPad, the image does not fill up the screen. The Apple TV still puts a pillarbox on the image, so I end up with a distorted, square image of my iPad screen. It's the right height - the max 600 pixels for my projector - but it's not as wide as it should/could be.

Is there a setting that I'm missing? I understand that the Apple TV has to stretch its home screen image to accommodate the 4:3 resolution, but I think it should be smart enough to realize that if it's receiving a 4:3 image from the iPad and sending it out to a 4:3 TV, it doesn't need to pillarbox the image.

Before you answer - this is not a problem with the projector settings, and my Apple TV is not defective. You could simulate what's happening on your Apple TV, even if it's hooked up to a widescreen TV. In the Apple TV settings, change the TV Resolution to 800x600 as mentioned above (or any other of the 4:3 options). Now go to the Apple TV home screen, and note the amount of space the image is taking up on your screen (on a widescreen TV, it will be pillarboxed, and it will look like it's been squished horizontally). Then turn on Airplay mirroring on the iPad, and you'll see that the iPad image does not have the same width as the Apple TV home screen and it looks square/squished.

Also, while my projector does have a zoom function, it will not magnify the small iPad mirror image correctly (if I were to leave the Apple TV and projector on a 16:9 setting). It cuts off the top and sides. But even if it didn't crop the image, it's a digital zoom, so that option still won't fix the fact that I'm not getting a full 800x600 image on the projector. The same problem would apply if I were to move the projector farther away from the screen (so that the widescreen image takes up the full height of the projector screen, while spilling off the sides). I'll still have a lower resolution image, which isn't ideal since I'm often showing text on the screen through Keynote. Also, widescreen videos wouldn't fit on the screen, and I can't move the projector easily because it's fixed to the ceiling.

Sorry for the long explanation. Any ideas? I'd really appreciate some insight.