The general circuit/formula minimization is certainly harder than identity testing, since the minimum formula size of any identity is simply zero. As for how much harder, I don't have a definitive answer but perhaps the "reconstruction algorithms" studied in arithmetic circuits/formulae might be something along these lines.
In these cases, you are give a blackbox and told that it is a formula in some class C (say a depth 3 circuit). The goal is to construct a representation of the blackbox in (something close to) C. Typically, most reconstruction results assume blackbox identity tests for the class, randomness, and sometimes other kinds of queries. Such reconstruction algorithms are available for certain restricted classes of circuits but not all classes for which we know blackbox PITs. Shpilka and Yehudayoff have a fantastic survey (pdf) on arithmetic circuits, and one of the chapters is entirely on reconstruction algorithms.
But in your case, you say d is a constant and hence even if the input was given as a blackbox, there are reconstruction algorithms for sparse polynomials. So maybe the above comments are not too interesting in this case.
Also, in the case of d=2, there are structure theorems for quadratics. Under a linear transformation on the variables, any quadratic can be rewritten in the form x1x2+x3x4+..+x2k−1x2k+ℓ. This property was used by Bogdanov and Viola for constructing PRGs for low degree polynomials (pdf) (Lemma 17 of their paper).