I you want to keep things simple, don't complicate the situation with a bursting balloon!
With the current system you have to worry about launching before the balloon bursts. A conventional "zero pressure" high altitude balloon is not sealed. It will achieve a higher altitude than a super pressure balloon* and maintain that altitude for long enough for a very relaxed launch window. If you prefer a sealed zero pressure high altitude balloon then, out of politeness, you may add a burster to deflate the balloon after launch.
*sealed latex weather balloons are "super pressure" balloons, because the internal pressure is higher ("super") than the external pressure. Zero pressure balloons have zero pressure differential.
A sealed zero pressure balloon is an envelope filled with enough gas to allow it to rise, but not enough to burst the envelope at altitude, but that means you need to fill it with a measured volume of gas. It's easier to partially fill a balloon with an escape vent at the bottom.
Welding a big polythene balloon is a reasonably simple d-i-y job, if you have access to a working area large enough to lay out your balloon segments.