you're making an app speciic association (which makes sense).  visibility has to do with what you can see in preview (eye icon) and/or render (image icon), not exporting.  if you look in your Wavefront Obj export settings, the first option should be  "Selection Only."  just select the item or items you want to export, and check that box.

the other settings are kind of useful, too.  i've got three export presets at present: one for props, one for morphs, and one for figures.  i've never actually used the default.

edit: oops!  x-post!