Does there exist a dye that will primarily bind with water in a water/oil emulsion?
You are asking for water-soluble dyes. Typical applications are dyeing methods in microscopy and food coloring.

Water-soluble dyes for microscopy: e.g. Eosin Y, Methylene Blue, Ethyl Green.

See also this table: Wikipedia in German: Lebensmittelfarbstoff - Deutschland.