Advice on facial animation with blendshapes AND bones

We have a project which needs a character to have a realistic animated face. In max the character is skinned to a standard biped which has an extra bone for the jaw. I want this bone so that I can add nice asymmetrical movement. The bulk of the facial expressions will come from morph targets/blend shapes however. The problem we are having is that we can't use Unity's in-built animation clip function because there's no way to scrub along with a sound file. But we can't do it in max because the morph target animations don't carry over with the fbx.The morph targets themselves do but not the animation I apply in the timeline.

So in summary - we can either do the jaw bone animation in max but no morphs coming in. Or I can try to animate the jaw and the morphs in Unity but without a sound file to work from. Which isn't practical.

How does anyone else do this? Presumably I'm not the only person to have come up against this?