How about
$$\frac{e^{-e^x}}{1+\epsilon x^2}$$
If you compute the Fourier transform you can shift the contour to height $\pm\pi/2$ to get an $e^{-|t|}$ times something decaying to 1, by Riemann-Lebesgue lemma

Edit.

Or you can look at a shift of your original example: $f(x+\log A)$ to get something on the lines of
$$e^{-A e^x} e^{-B x^2} e^{C x}$$