We are working with Maya 2009 for our current project and our game uses FBX format for assets.
I -need- to use blend shapes for one of our characters, and while I can bake the blend animations into the final export, the blend models remain in the exported file, resulting in 6 sets of eyes and a bunch of mouths.
Is there an alternative animation method or a way to remove the blend models before exporting so that I can have all the wonderful character expression that blends provide without having spare parts floating around?
Just toggle the renderer on those. They won’t render and thus won’t make your project heavier 
It seems blends just won’t export properly at all. Is there an alternative? Lattice’ seem buggy as well, and doing facial expressions with bones seems like it could get messy.
Oh you’re talking about importing in unity, i thought the only problem was visualizing them because you were using the script from the wiki.
Blendshapes are not supported in unity, as is not every other per-vertex animation (lattice for one).
There is a script on the wiki for doing blendshapes, check that out