Ignoring mistakes such as the lack of a window title in alt+tab, messed-up program icon, and a restore button in the title bar that does nothing …
I do have to wonder if Switch Hitter ignores keys that the OS doesn't recognise, or whether Windows itself simply drops the requests at a lower level than Switch Hitter is running at, especially since Switch Hitter doe. I connected my Tactile Pro 3, and Switch Hitter recognises volume up, down and mute, but it doesn't register the eject key. Otherwise it behaves as a perfectly ordinary Windows keyboard, and § produces ` which is what's printed on the key in my layout. F14 is scroll lock, for example.
You say that you have 6 keyboards that do this — are any of them full size? If you've got a full-size one (AEK/AEK II/AppleDesign) with F13–F15, do those three register as print screen/scroll lock/pause as on a PC keyboard?