Can we find a function $\sigma\in C_c^\infty(\mathbb R)$ with $\sigma(\tau)=1$ for $|\tau|\leq 1$, $\sigma(\tau)=0$ for $|\tau|>2$, and $$ \int|\widehat{\sigma^{(n)}}|\leq 2^n $$ for each $n\geq 0$?
I am asking this because I need to find such functions for a proposition in my thesis.
No, this is not possible.
From Sobolev embeddings (or fundamental theorem of calculus) and your integral condition one would have for every $x$ $$ |\sigma^{(n-1)}(x)| \leq C^{n} $$ and so $\sigma$ is analytic, and so cannot be compactly supported.