Once you know that the support of $\hat T$ is $\{0\}$, you are done: in fact, $\hat T$ is a (finite) linear combination of derivatives of the Dirac mass at 0, whose inverse Fourier transforms are monomials.

@Carlo Beenakker Thanks for this sharp calculation. With $$ F_{k}(x)=\int_{\mathbb R}\frac{\sin x t}{\pi t}\frac{(1+it)^{2k+1}}{(1+t^{2})^{k+1}} dt, $$ we get indeed $F_0(x)=1-e^{-x}$(your calculation), and also $F_1(x)=1-e^{-x}(1+2x)$, $F_2(x)=1-e^{-x}(1+2x^2)$, $F_3(x)=1-e^{-x}(1+2x-2x^2+4x^3/3),$ and a general formula involving the Laguerre polynomials.

The rhs of the first equation is simply the bracket of duality between the distribution $\Gamma$ and the smooth compactly supported function $(x,y)\mapsto \psi(x)\phi(y)$.