Maybe we could use something like the Cotton Candy. Basically it is a little computer running Android or Ubuntu in a USB stick, with its own wireless network connection, that you can plug into a pc. With some client software on a Mac, PC or Linux, it will essentially take over the host computer's screen, keyboard and mouse, with all the software running on the stick. Or you can run it off a TV with a USB or Bluetooth mouse or PC.
At testing time, HQ would ship out the testing sticks to schools, with as much of the bandwidth-intensive media files as might be needed by the tests already loaded to cut down on network requirements. Every student could have an identical, secure testing environment regardless of the vagaries of the local PC stock.
Presumably people paid to think about this kind of thing are already doing so. I don't know.