Thank you, oliver and Mogensen, for your info !
Now I know why the lady at the EK office couldn't issue the layover voucher without the passport (or copy) of my girlfriend in hand. She said that I could collect it on the day of my trip while in transit at CAI's T2 at an EK desk airside (the landside office is closed). Probably I will try to get the voucher ahead of my trip by contacting FRA's EK office (hope they will cooperate).