The Berry-Esseen theorem's statement involves random variables who have finite first three moments. But it seems like your random variable doesn't even have a variance... If you mean for the $\log\log$ term in the denominator to be a correction for this, then the distributions the statistic $S_n$ works on now vary with $n$. How is this going to lead to a counterexample?

The hard part is making sense of what a conditional probability distribution means math.stackexchange.com/questions/496608/…

This is discussed, at least in part in arxiv.org/pdf/1709.06522.pdf where they look at deviation of these polytopes from spherical caps.

Is Analysis Now by Pedersen one of the books you consider to have a superficial treatment? If it is to the appropriate level you might specifically be interested in the last chapter: $${}$$ > This chapter has two functions: Throughout the book it has served as an Appendix, to which the reader was referred for definitions, arguments and results about measures and integrals. It will now serve as a functional analyst's dream of the ideal short course in measure theory. $${}$$ It's placed after a rigorous development of spectra by Gelfand transform and an overview of unbounded operators.

If you are willing to scale $Y(t)$ so that it's power is 1, Lemma 1 (then Lemma 2 and the rest of the paper) might help, as it turns bounding $S(Y(t)/\sqrt{\langle Y(t)^2\rangle})$ into bounding two mutual informations. sciencedirect.com/science/article/pii/S0019995878904138

What sort of probabilistically meaningful information do you hope such a distance metric will describe? Maybe I am being cynical but I can't see how it would preserve much... you can easily design distributions that are distant in your sense [up to some interpretation] but close in many other very strong ways.

One immediate (maybe naive) reason these objects are undesirable to study is that your class of stopping times becomes greatly restricted: if your random walk is non-causal then you can no longer decide when to stop based on the walk's history. What is gained by interpreting its index as temporal? Maybe what you have in mind is an object like a Brownian sheet, or some process over a non-linearly-ordered domain. But I would not call such a thing a 'random walk'.