How can I use the Banach fixed-point Theorem to find approximately the Omega Constant?

Question

I know how to find the Omega constant using Newton's method, however, I was trying to compute the Omega constante $$\Omega \approx 0, 567143290409783872999968662210 ...$$ by using Banach Fixed-Point Theorem in the equation $x=e^{-x}$. As far as I know, we should define some $T(x) = e^{-x}$ then prove that $T(x)$ is a contraction. Then, we Shold iterate that contraction in order to find the constant. However, I cannot prove that $T(x)$ is a contraction and doing the iteration I do not find the Omega constant.