Let $g_n, \phi \in \dot{H}^1(\mathbb R^d)= \{f: \nabla f \in L^{2}(\mathbb R^d) \}.$
Assume that $\int_{\mathbb R^d} \nabla g_n \cdot \nabla h \ dx \to \int_{\mathbb R^d} \nabla \phi \cdot \nabla h \ dx$ as $n\to \infty$ for all $h\in \dot{H}^1.$ (In other words, $g_n$ converges to $\phi$ weakly in $\dot{H}^1$.)
Question: Can we expect to find $\psi \in \mathcal{S}(\mathbb R^d)$ (Schwartz Space) so that $$\lim_{n\to \infty} \int_{\mathbb R^d} g_{n}(x) \psi(x) dx = \int_{\mathbb R^d} \phi(x) \psi (x)dx?$$
Side Note: Let $\phi$ be a radial bump function supported on $\{ \xi: |\xi|\leq 2 \}$ which equals to 1 on $\{\xi: |\xi|\leq 1\}.$ Put $h(\xi)= \phi(\xi)- \phi(2\xi),$ and $h_{1}(\xi)= h(\xi/2).$ Take $\psi(x)= (h_{1})^{\vee}(x)$ (Inverse Fourier transform of $h_1$) (Is this the right choice?)
One issue is your precise definition of $\dot {H}^1$. You need to factor out the constants, i.e., it is a subspace of $\mathcal {S}'/\Bbb {C} $.
Anyway, you have $g_n \to \phi $ weakly, i.e., you only need to verify that $g \mapsto \int g \psi $ is a linear bounded functional on $\dot {H}^1$. But now, if $\psi$ is a Schwartz function such that $\hat {\psi} $ is supported away from $0$, then $$ \left|\int g \psi \right|=| \langle g, \overline {\psi} \rangle_{L^2}| = |\langle \hat {g} \hat {\overline {\psi}}\rangle_{L^2}| \leq \int |\hat {\nabla g}(\xi)| \cdot |\overline { \hat{\psi}(-\xi)} |/(2\pi |\xi|) d\xi $$ can be estimated by a constant multiple of $\|\hat {\nabla g}\|_{L^2} = \|g\|_{\dot {H}^1} $.
Above, I repeatedly used Plancherel''s Theorem. Also, the last line might be slightly different depending on the normalisation that your are using for the Fourier transform (give or take a factor of $2\pi $).