You need a different approach. Each function in your function space can be written as
$$F_{Y|W}(y|W) = \int 1(s \leq y) P(Y = ds|W)$$
for some $y$. Thus,
$$\|F_{Y|W}(y_2|W) - F_{Y|W}(y_1|W)\|_{L^1} = E_{P_W}|F_{Y|W}(y_2|W) - F_{Y|W}(y_1|W)| \leq E_{P_W}\int 1(y_1 \leq s \leq y_2) P(Y = ds|W) \lessapprox \|1(Y \leq y_2) - 1(Y \leq y_1)\|_{L^1}.$$
A similar argument probably works for $L^2$.
In particular, your function space is obtained by a Lipschitz transformation applied to the function space
$$\mathcal{F}_{ind} = \{s \mapsto 1(s\leq y): y \in \mathbb{R}\}.$$
So your covering number is essentially that of $\mathcal{F}_{ind} $.