The best way is to make a small payload, and put it on a long long line to the balloon and let the laws of physics ( and the low frequency of the pendulum you're left with) do the rest.
Doug, having just looked through the videos again:
The images are lovely - though I'm trying to work out whether the drop in pitch of the on-board mic with altitude is a function of air pressure or cold or both.
A more stable instrument platform would seem to be the next step. As you say, a longer tether would lower the frequency. It would also help by moving the package out of the worst of the turbulence "behind" what is an unstreamlined object moving at a fast jog (16 kmph) straight up.
Since vanes to supress horizontal rotation can't work (surely the balloon is moving virtually at any surrounding horizontal airspeed?) and would be a disadvantage under pedulum swing (they'd feathervane the package on each swing), why not try increasing the rotational inertia of the package?
If you went for a pair of long booms with a mass at the ends, it'd be more stable - though still free to move, allowing for panoramic views. Better, if you had three booms, you could go for a long tripod-like tether which would hang the package with a virtually assured horizon. Mass increases could be minimal - and you could boom-mount a camera with no negative effects.
Placing the parachute below the balloon but above a bridle for this sort of tether would surely not increase the risk of tangle? (In the video we get a quick shot of near zero g when the package starts falling with balloon material around it, until the parachute fills.)
Incidentally, tell me the parachute falling juicily into frame in the last video is a fix?! Too lucky by half!!!