Slowest decay for Fourier transform of test functions.
Let $\phi = e^{-1/(1-x^2)}1_{|x|< 1}$. Its Fourier transform is real and even.
For any $\hat{g}\ge 0$ continuous even decreasing on $[0,\infty)$ and such that $\forall n, \hat{g}=o(t^{-n})$
Then $$\hat{f}=\hat{g} \ast \hat{\phi}^2$$ doesn't decay faster than $\hat{g}$.
And its inverse Fourier transform $$f=g \cdot (\phi \ast \phi)$$ is $C^\infty_c$, since $g$ is $C^\infty$ and $\phi \ast \phi$ is $C^\infty_c$.