Why do I have brown, smelly discharge coming from my belly button almost a year after my tummy tuck?

You need to get second opinions. The source of the drainage needs identification. The drainage needs lab analysis.  You may require exploration around your belly button to help solve the issue. 

I agree with my colleagues that you likely have an infection from some of the internal sutures that were used during your tummy tuck. You may also have a collection of fluid (seroma) that has become infected and if leaking out of an opening around your belly button. Either way, you should see your plastic surgeon. If you are unhappy with your plastic surgeon, you can find another one to help you - keep in mind that you will need the prior operative report so that a new surgeon will know what the original surgeon did (this will help him/her treat you.)

Occasionally the navel can retract and create a sinus that will be perpetually draining in foul-smelling. This would require a relatively trivial revision.