@Lelouch Net PE of the system (rod+balloon) is not zero because you can convert it into to kinetic energy by slightly tipping the rod and thus causing it to rotate. If initially the air density inside the balloon were identical to that of ambient this would not be possible, so the initial configuration has zero PE.

@lemon Mass doesn't change but effective gravity acting on the two balloons changes when their densities change (think buoyancy). If it is any help, you may think that initially, density of air in both the balloons was identical to that of atmospheric air and so there are no buoyancy forces on them (equivalently effective gravity on them is zero).

@LeLouch You obtain a top-heavy configuration in the latter case.