For posterity, after talking with several product specialists, zinc picolinate does not qualify as zinc being bound to an amino acid. It's bound to picolinic acid, which is just another organic acid. Zinc citrate and copper gluconate also do not qualify.
I did find a cheap source of amino acid-bound copper. It's by Puritan Pride's and the main ingredient is copper amino acid chelate. Upon inquiry, I discovered the amino acid was glycine.