3-Coloring Problem Question
First, it is not known that 3-coloring takes exponential time. Even if P is different from NP, it could still be the case that 3-coloring can be solved in subexponential time, though the Exponential Time Hypothesis (ETH) precludes this possibility (if you assume the hypothesis).

Second, regarding your suggested solution. The black box only accepts graphs, it doesn't accept partially colored graphs. You need to give a reduction which only feeds the black box  actual graphs.

Third, if a graph is 3-colorable, then there is always a 3-coloring in which a particular vertex is red. So what you're really trying to do is ask whether there is a 3-coloring in which a given set of vertices has the same color, another set of the vertices has a different color, and yet another one has the remaining color. See if you can somehow encode some of these constraints using graphs rather than partial colorings.