the problem has been isolated to the basement floor drain going to the main City storm drain.
So, when the City drain gets overwhelmed it backs up through your basement pipe then out your floor drain????
Have you had this drain in your basement and the pipe reamed out for cleaning and stuff??
If this pipe is clear and OK, then I would have to say the problem is with you and the City. And since this only happens with torential rains of 5 to 10 inches in magnitude, this just might be an act of God. Meaning there is not remedy.