As you mentioned, what they say is only true if $N = N(\omega)$ is itself random. However, one could apply Egorov's theorem to get almost sure uniform convergence on an event that occurs with arbitrarily high probability. In which case, one could grab a nonrandom $N$ such that their statement holds with arbitrarily high probability. If they are only making $O_P$ statements, this might be enough for the argument to go through.

For the above, I took $\|\cdot\|_p$ as the norm in $L^p(K)$, but you might not want this.

Assume $|f(x) - f(y)| \geq L |x-y|$ uniformly then $|f_{\theta}(x) - f_{\theta'}(x)| = |f(x-\theta) - f(x-\theta')| \geq L|\theta - \theta'|. $ In particular, $\|f_{\theta}- f_{\theta'} \|_p \geq L|\theta - \theta'| diam(K)^{1/p}$. So a lower bound would be $L diam(K)^{1/p}$. I don't know how useful this is for you as it pretty much limits $f$ to being strictly monotone with derivative uniformly bounded from $0$.

I have added an edit to my response. I believe corollary 23 of the referenced paper states that any continuous function of finite quadratic variation with only countably many nondifferentiable points has quadratic variation zero. If this is indeed what it says then the answer to this question is affirmative.

$F_{Y|W}$ is the same for all functions in your space. So your convex hull argument is not correct (or at least does not allow you to handle the $W$ randomness).

Also, your function space is a bit weird. Are you treating $W \mapsto F_{Y|W}(y|X)$ as an element of your space and then varying over $y$? In this case, you need a different approach.

See this paper and references therein for the covering numbers for CDF's of signed measures: arxiv.org/abs/1907.09244.