The notations are not good. For example, when you write $$\int_{X\times A}\; \sup_{f\in \mathcal{F}}E \big[u\big(x,f(a)\big) \big| \mathcal{A}\big] d\pi(x,a),$$ $u(x,f(a))$ in the integral behaves like a constant so $E \big[u\big(x,f(a)\big) \big| \mathcal{A}\big] = u(x,f(a))$.
You should write instead $$\int_{X\times A}\; \sup_{f\in \mathcal{F}} E \big[U_f \big| \mathcal{S}\big] d\pi,$$ where $U_f$ the random variable defined bu $U_f(x,a) = u(x,f(a))$ and $\mathcal{S} = \{\emptyset,X\} \times \mathcal{A}$.
Anyway, the equality cannot be always true. For example, consider $X=A=\mathbb{U}$, unit circle in $\mathbb{C}$. Endow $X \times A = \mathbb{U}^2$ with its Borel $\sigma$-field and $\eta \otimes \eta$, where $\eta$ is the Haar measure on $\mathbb{U}$. Then taking conditional expectation with regard to $\mathcal{S}$ is just taking averages over the fist component.
Let $\mathcal{F}$ be the family of all rotations on $\mathbb{U}$, and $u : (x,y) \mapsto \Re(xy)$. Then for each $f \in \mathcal{F}$, the average over $x$ of $U(x,f(a)) = \Re(xf(a))$ is $0$. Yet, since $$\sup_{f \in \mathcal F} U(x,f(a)) = \sup_{y \in \mathbb U} \Re(xy) = |x| = 1,$$ the average over $x$ of $\sup_{f \in \mathcal F} U(x,f(a))$ is $1$.