Clearing the balloon isn't the problem...
There are a couple of ways to do this...
1) connect the aircraft to the balloon at the center of its mass. When you drop it, it should go in to a normal, 'flat' glide path. Delay, ignite the engine and then after a second delay you alter the control surfaces in to a climb. You should be clear of the balloon. (Someone says that the kit to do this exists already.)
2) You drop the aircraft with a weighted tail so that the aircraft drops tail first. (like a tail first stall.)
After a calculated delay, you start the engines which also removes the excess weight on the tail.
(Dry ice maybe?) This should launch the aircraft at a high angle but it should miss the balloon.
3) Look back at some physics of a rotating arm and the atlatl. At launch the arm and aircraft are parallel to each other hanging straight down. At launch, the arm swings the aircraft on an arc. You can calculate the length of the arm based on the size of the balloon. You can determine the launch point and use a small explosive bolt to release the plane from the launch platform. The engine ignites some time after detaching from the launch arm.
The key to #3 is that you need to have a strong flexible arm. My suggestion is a graphite pole like you see in a golf club. You also need an explosive bolt to aide in the release. And you'd have to deal with the timing of the release. (mercury switch or something...)
This allows you to have 1 balloon and a relatively simple launch platform. Also easier to test the components like the detaching bolt, and also ignition tests on the engine. You could also preset the launch altitude too.
But what do I know?
I'm not a rocket scientist. :-)