Do you drive it daily?
Driving or Charging should charge the 12v Battery.
So, I you drive most days, it sounds like the 12v Battery is not getting charged,
or there is an unusual current drain. The dealer should be able to easily measure
the current drain without the "keys" in the car, and see if there is an
abnormal drain. Then, in the same test, toss the keys in and see what happens
to the current.
I would think that the RFID overnight, or for a day, would
not be a problem, but... try not doing that.