This idea took the most thought to implement.  The Barbies needed to be suspended for the jumps and cartwheels.  I built a stand with two threaded rods.  The Barbies were connected to the rods with an eye screw and wire.  This gave them some clearance from the rod and suspended them at the waist.  That gave them all the freedom they needed.

The actual animation went rather smoothly.  For the second half of the clip, the girls could follow their routine on video to time the Barbie animation.  Barbies are not very articulated.  They cannot move their elbows or knees (much), which made some of their maneuvers awkward - but what little girl doesn’t want to cheer with Barbie...