I stumbled on the following problem, if you can see a way through it.
Let $x$ be a real variable and fix a real value $\frac14\leq\nu\leq\frac34$.
QUESTION. For $x\rightarrow0$, does there exist a constant $C>0$ (independent of $x$ that is) with the below property? $$\int_{\mathbb{R}}\left\{\int_0^{\infty} \left(e^{-\frac12\left[\frac{x^2(1+y)^2}{z^2}+z\right]} -e^{-\frac12\left[\frac{x^2y^2}{z^2}+z\right]} \right)z^{-\nu}dz\right\}^2dy \,\,\,\,\sim \,\,\,C\,\vert x\vert.$$
Replacing $y$ with $y-\frac 12$, we get the integral $$ 2\int_0^\infty\left[\int_0^\infty z^{-\nu}e^{-\frac z2}e^{-x^2\frac{y^2+0.25}{2z^2}} \left(e^{\frac{x^2y}{2z^2}}-e^{-\frac{x^2y}{2z^2}}\right)dz\right]^2\,dy $$ Now, for small $x>0$, reduce the integration to $x\le z\le 2x$, $1\le y\le 2$. Then the inner integral is about $x^{1-\nu}$ and the final answer is at least $x^{2-2\nu}$, which is much larger than $x$ for $\nu>\frac 12$. Off hand, it looks like the region I described gives the main contribution in all cases but I haven't checked it.
