= Solution
At a site with background $\eta_i\widetilde{\mathbf n}$, the linear variation is $\delta\mathbf n_i=\mathbf m$. The supplied variation formula gives
$$
S\,\delta\Gamma[eta_i\widetilde{\mathbf n}]
=S\int dt\,mathbf m\mathbin\cdot
\left[(\eta_i\widetilde{\mathbf n})
\times(\eta_i\partial_t\widetilde{\mathbf n})\right]
=S\int dt\,mathbf m\mathbin\cdot
(\widetilde{\mathbf n}\times\partial_t\widetilde{\mathbf n}).
$$
Replacing the lattice sum by $a^{-2}\int d^2x$ yields
$$
\boxed{
I_{WZ}^{(1)}
=\frac S{a^2}\int dt\,d^2x\,
\mathbf m\mathbin\cdot
(\widetilde{\mathbf n}\times\partial_t\widetilde{\mathbf n}).}
$$
This <Berry-phase term> makes the uniform canting field the momentum conjugate to rotations of the Néel order parameter.
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