I have personal experience with a roll over accident. I rolled my Mustang in 1995 and the only think that saved me (in addition to my seat belt, of course) as I rolled off the highway doing 65 mph, was that I was driving a hard top and not a convertible. The entire roof of the car was dented and misshapen. After that experience, in which my scalp was very sore and bright red from hitting the roof of the car as the vehicle rolled down an embankment, I would never ride in a convertible myself, unless it had a roll bar, let alone with my kids. I guess I wouldn't think he would be more likely to get thrown from the car, but to be crushed in the accident.
I hear what others are saying about convertibles being bottom heavy, but honestly, how much could the top of a car weigh that it would make any difference?
I don't think you are being too paranoid at all. Stick to your guns! ;)