Apple TV / iPad / Projector aspect ratio issue

I work in a school which has projectors and screens in most of the classrooms and we've just started installing Apple TVs. I’ve come across a problem with using AirPlay to mirror the iPads using the Apple TV connected to a projector. All our projectors are 4:3 aspect ratio, either SVGA (800x600) or XGA (1024x768). The Apple TVs are 16:9 by default so are letterboxed when displayed on the projectors.

4:3 projector screen:

16:9 Apple TV on 4:3 projector screen:

If I then mirror an iPad which has a 4:3 display via the Apple TV, it in turn gets ‘pillarboxed’ so the iPad image is small and doesn’t fill the whole screen:

I've tried manually selecting a 4:3 resolution from the Apple TV settings, but the image is just squashed and looks like this:

I can’t find a setting on the Apple TV to tell it that it is connected to a 4:3 ratio display and so fill the entire screen.

However, if I don't mirror the iPad's display but send a video from, say, the YouTube app, the image fills the entire 4:3 screen.

Any ideas would be gratefully received.