Right, let's speculate!
You go up with the balloon, then want to use the gas for fuel. This will require several things. We need a balloon that can be deflated on command (plausible) , a controlled burn gaseous hydrogen rocket (difficult) and a main capsule that can control its orientation in the upper atmosphere.
Initial thoughts would be to launch the balloon up to a certain height, then force deflate it, compressing the gas internally within the main capsule. During this stage the craft will be in freefall, so the gas needs to be compressed as quickly as possible to minimise altitude loss. Problem: Heat from compressing the gas could lead to explosion, need study to find out how likely this is.
Once the hydrogen is compressed, we dump the balloon (with a parachute for recovery, balloons ain't cheap) align the main capsule and hit the burn button. This ignites our carefully released hydrogen fuel (some form of pulse rocket? Need to think about that.) and we burn for LEO. Do the rules specify what they're considering "space" to be? Anyway, once up, we release the satellite (Electronics chappies can probably do something with 20g) and hope it's somewhere vaguely stable for the next ~12 hours.
Recovery of the main capsule is a non-trivial problem left to the reader.