In a finite abelian group, I think we have the relation

f\*g = sum over pairs (x,y) in the group of ([y]\*([x].f)).([x]\*g)

Here [x] denotes the indicator function that takes the value 1 at x and zero elsewhere.  On the trivial group this says f\*g = f.g, so I don't think it reduces to 0 = 0.  The relation can be deduced from the fact that ([u]\*h)(v) = h(v-u) for all h, u, and v, and that summing the terms over just y gives f(x).([x]\*g) = f(x).g(blank - x), and the definition of convolution.  Maybe this is cheating but it follows the letter of your criteria.