It's always helpful if you can put what you have in a simple demo. It doesn't have to be pretty, or even working. Just something we can experiment with.
For this, I would probably morph the lips as a mask or clip path for a group element (<g>). In that group, I would put a darkblue rectangle to give it a background color, and for the mouth, well it probably doesn't need to be morphed. The mouth shape could probably be left as is, but move it down on the y-axis so it's not visible in the starting state. You could also scale it down a bit if needed.
And to animate it, just morph the mask or clip path while moving the mouth up. But that's just one way to do it. There is no correct way.
And animating masks and clip paths have their own little quirks, so you should test to see what works best. If you haven't already, check out the SVG Gotchas thread. It has some good tips on working with masks and clip paths.