= Solution
The equal-time <canonical commutation relation> for the transverse-traceless graviton and its <canonical momentum> is the <transverse-traceless projector>. At zero momentum the supplied <polarization completeness relation> gives
$$
i[Q_S,\gamma_{mn}]
=-2\omega_{ij}\sum_s2\epsilon_{ij}^s(\mathbf0)
\epsilon_{mn}^s(-\mathbf0)
=-2\omega_{mn},
$$
where symmetry and tracelessness of $\omega_{ij}$ remove the trace term. Thus the charge generates precisely the transformation in part i:
$$
\boxed{i[Q_S,\gamma_{mn}]=\Delta\gamma_{mn}}.
$$
This is the soft, field-independent part of the <Noether charge> associated with the large diffeomorphism.
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