Batteries in FOB's are funny things. I had a similar situation with my 2002 VW TDI. The immobilizer started tripping not allowing the car to start. Tried several things and finally changed the FOB battery. It solved the problem even though the FOB worked the remote locks perfectly with the old battery. Go figure

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2012 Mercedes ML350 Bluetec 91K (hers)
2005 Corvette 55K (fun car)
2002 VW Jetta TDI 231K (mine)
1998 Volvo S70 T5 Turbo 196K (kids)
1994 Ford F150 4WD 249K (firewood hauler)
1983 Mercedes 300D 376K (diesel commuter)