I swapped the battery from the Bird (480 CCA) to the Corvette's battery (540 CCA). I made a test in the garage. Of course the cold engine started fine as expected. I heated it up and stopped the engine. And … it immediatly started again without any hesitation. To countercheck, I swapped the battery again to the 540 CCA and the old phenomenom was back.
Amazingly the car idled in warm condition and the Vette's battery without any problems. I always thought, the battery does not play any role when the engine is running. There must be some influence.
At the moment it seems, that the battery solves all of the problems. I will do a road test as soon as possible (without all the salt on the streets).
@Gus: I am an old C64-guy and still have 2 of it! ;-)