Is there a way to send data back to the Teensy? That way you could have a program on your computer that tells it "hey, I'm in vim now, so use 'i' and 'Ctrl-['" or "hey, I'm in emacs now, so just be 'Left-Alt' now".
If it's "just a keyboard" and doesn't normally support that, I've seen hacks that utilize the Caps/Scroll/Num lights as inputs back into a keyboard...
I'm not sure if there is a way to send signals back to the device. But, it should be trivial to do what you describe on the operating system level. Ignoring the possibility of specifically looking for that keyboard, you could just have is send obscure keys that you would never type. Then you can just re-map them on the OS level whenever you want to.
If it's "just a keyboard" and doesn't normally support that, I've seen hacks that utilize the Caps/Scroll/Num lights as inputs back into a keyboard...