I was looking at the beads and thinking, since they have such tiny holes (as all gorgeous stone beads do, it seems), another option could be to run the fine wire though the bead and create a small bail, then use a larger gauge wire to connect the bead bail with the finding and wrap over the smaller bail/bead.
Maybe another thing to try is to wrap the end wire on one or both sides of the "triangle" wire that runs up from the bead hole.