Help with a zero knowledge proof
It could be straight-forward with verification equation quadratic in challenge, compared to well-known equation of Schnorr protocol that is linear in challenge.

In particular, for well-known responses like
$X = xc + \alpha$, $Y = yc + \beta$, $R, R', Z$
and equations for $C_1, C_2$ do it:
$(g^{R'} h_1^X h_2^Y)^Z = C_3^{c^2} D_1^c D_0$

Produce explicit formulae for $D_1$ and $D_0$ to complete this homework.

$(g^{r' c + \delta} h_1^{x c + \alpha} h_2^{y c + \beta})^{z c + \gamma} = C_3^{c^2} D_1^c D_0$

This approach with higher-degree polynomials in challenge rely on an upper bound for number of roots, two in this particular case.