Is it a "real" car key, with a metal bit you put into the ignition & turn?
If so, you would be better off forgetting any "high tech" methods related to its RFID function, & just look for it with a metal detector.
They are usually sensitive enough to detect wedding rings & the like, so they should be able to pick up the key.
In a pinch, the copper on the PCB may be detectable.