Try the Lufthansa car pool. CX staff can rent cars from them for as long as they like x FRA and Berlin.
LH staff drive in during the early morning and drop the car, then you simply select which one you want, tell them where in Europe you will be going ( so they can insure you etc ) and off you go.
There office opens about 0700 in the carpark. A very small window to their office beside a security exit one way door. ( which we cannot enter without local ID cards ) is the only way non- LH staff can speak to the staff and do the paperwork. Just wait and someone will come to the window.
DON'T FORGET YOUR CX ID.
Not sure of their phone number to check but I'm sure someone in CX FRA would know.
Edit:- there appears to be a number on the top of the rental rates form for Frankfurt. I used them about 4 years ago and seem to recall ringing them beforehand to check. They said yes we allow CX staff to take the cars no problems.
Here is the link .
Lufthansa Carpool