I don't think his C43 has that alarm function. (red/green lights in the rear view mirror)

the 113 V8's had a problem with the crank shaft position sensors. Usual symtpoms were from hot engines -when idling at high temps vehicle would shudder then quit, or when vehicle was hot, turned off and then would not restart. If the vehicle starts fine when cold, runs great, and you have no codes -you may need to start with a crank sensor.
BEST GUESS for me.

The Key batteries have nothing to do with starting, only remote functions. You can take the batteries out and still start the car.