Yes, yes, for every action there is an equal and opposite reaction. But, if the launch platform also had engines that launched at the same time as the rocket, they could counter the downward thrust. also, rocket propulsion in the atmosphere actually uses the atmosphere as if it were an opposing force. Put a rocket horizontally on a clothes line and set it off and it will move forward yet the only opposing force it has is the air itself. Additionally, the rockets initially force is so quick and the rockets weight so light that the mass of the platform would not instantly drop at an equivalent speed of the total upward force of the rocket, even without counter propulsion. Some speed would be knocked off that it would have on the ground, but not a significant amount as to cause the platform to drop instead of the rocket lifting off. Float a foil covered Styrofoam platform of say 4'x4' in a pond of water with a rocket on it and the rocket will launch but the platform won't sink because the rocket is aerodynamic while the pressure required to submerse the Styrofoam is significantly greater than the brief thrust coming from the model rocket.
In short, I think it could be done with a triangular platform of ultra lightweight material like balsa with three balloons attacked, one to each corner, and the tops of the balloons attached together such that the move as one. The rocket could be put at the center of the triangle and launched from the space between the balloons. Additional thrust could be put into the rocket and not lost to the platform by attaching a second identical rocket at the center point underneath the platform that fired at the same time, thereby countering some of the lost thrust do to the brief platform descent.