We eventually sorted this out, and it appears as (one direction of) Theorem 3.6.7. in [*Dualizable Tensor Categories*][1] (joint with Christopher Douglas and Chris Schommer-Pries).  Note that (for C semisimple) separability is equivalent to semisimplicity of Z(C) (see Corollary 3.5.9.), so Theorem 3.6.7 does address exactly this question.


  [1]: http://arxiv.org/abs/1312.7188