= Solution
First interpret the displayed critical <correlation function> as the <connected correlation function>
$$
G_c(x)=\langle m(x)m(0)\rangle-\langle m\rangle^2.
$$
Above the transition at zero field the subtraction is zero; in an ordered phase it is essential. For a dimensionless source coupled to the total <magnetization> $M=\int d^dx\,m(x)$, differentiating $Z$ twice gives $\chi=V^{-1}(\langle M^2\rangle-\langle M\rangle^2)$. Translation invariance then proves the <correlation-function susceptibility sum rule>
$$
\chi=\int d^dx\,G_c(x).
$$
Insert the scaling form, use spherical coordinates and set $s=r/\xi$:
$$
\chi_s=S_d\int_a^\infty dr\,r^{1-\eta}g(r/\xi)
=S_d\xi^{2-\eta}\int_{a/\xi}^\infty ds\,s^{1-\eta}g(s).
$$
For $\eta<2$, finite nonzero $g(0)$ and an integrable large-$s$ tail, the last integral tends to a finite nonzero constant. Microscopic separations add an analytic response background. Thus the singular <magnetic susceptibility> obeys $\chi_s\asymp\xi^{2-\eta}$, and at zero field $\xi\asymp|t|^{-\nu}$ gives the <Fisher scaling relation>
$$
\boxed{\gamma=(2-\eta)\nu.}
$$
A dimensional magnetic field would add a nonsingular inverse-temperature prefactor. Integrating the full unsubtracted ordered-phase <correlation function> instead would yield a volume-divergent term $V\langle m\rangle^2$, which is not the intrinsic susceptibility appearing in this exponent identity.
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