So close...
\begin{align}
& \langle(0e^{-y^2}+e^{-x^2})*\delta^2(x,y),\phi\rangle=\underbrace{\int_{\mathbb{R}}\cdots\int_{\mathbb{R}}}_{4}(0e^{-y'^2}+e^{-x'^2})\delta(x)\delta(y)\phi(x-x',y-y')dx'dy'dxdy
\\
& =\underbrace{\int_{\mathbb{R}}\cdots\int_{\mathbb{R}}}_{2}(0e^{-y'^2}+e^{-x'^2})\phi(-x',-y')dx'dy'
\\
& =\int_{\mathbb{R}}e^{-x'^2}\phi(-x',-y')dx'
\\
& =(e^{-(\cdot)^2}*\phi (\cdot,y))(x)
\end{align}
But
$$\mathcal{F}(0e^{-y^2}+e^{-x^2})\cdot \mathcal{F}(\delta^2(x,y))=e^{-x^2}$$
Surely multiply any test function by a gaussian is not the same as convolving a gaussian to the same…