Under the assumption that $g_n$ are nonnegative bounded functions $\mathbf{R} \to \mathbf{R}$ such that for all $n$: $\int_\mathbf{R} g_n = S$, and $g_n \to 0$ uniformly on any compact set $K \subset \mathbf{R}_{\neq 0}$, I'm trying to show that the following statement holds
For all $\epsilon > 0$, There is some $N$ such that $\left| \int_\mathbf{R} g_n f - S f(0)\right| < \epsilon$ when $n > N$.
In the statement above $f$ is rapidly decreasing (aka, Schwartz). Any hints on getting an estimate?
That is not true. You probably think that your conditions force the mass to be shifted to the origin, but it can as well be shifted to infinity. Here is an example:
Let $g_n=1_{[n,n+1]}$. Clearly, $\int_{\mathbb{R}} g_n=1$ for all $n\in \mathbb{R}$ and $g_n\to 0$ uniformly on compact sets. However, $\int_{\mathbb{R}} f g_n\to 0$ for all Schwartz functions $f$ (or indeed all $f\in L^1(\mathbb{R})$) by dominated convergence. Indeed, this will always happen if the sequence $(g_n)$ is uniformly bounded.