The battery in the FOB is for the transmitter that locks and unlocks the doors, and only that.
The RFID chip in the key is what enables engine start, but once the car is running, it's job is over.

This is a classic "my Crank sensor won't let the car run until it cools off...."

Go get yourself a BOSCH crank sensor at GermanAutoSupply.com or FCPeuro.com.

YOu can buy one at Advance or where ever, but often their parts are not quite to spec and you end up fighting a ghost.

Just go online and google "Mercedes 2003 SLK320 Crank Position Sensor" and buy the BOSCH part, not the cheap fake parts.