So, overall, we generally don't have much of any evidence of any biblical legal material being carried out on the ground. I think it's important to remember that legal codes as they existed in the ancient Near East (and here I'm thinking primarily about the Bible and Hammurabi, but I'm sure others like the material from Hati would be similar) were largely academic documents. To be sure, there are many ways in which they reflect cultural proclivities on a variety of issues—dietary, interpersonal relationships, perspectives on authority, etc. However, we would really need more textual data from outside the biblical text to corroborate what laws may or may not have been enforced. One such example would be the Meṣad Hašavyahu ostracon, in which a worker had his garment taken in pledge and not returned in a timely manner (i.e., as collateral; see Exod 22:26; Deut 24:17; Prov 20:16, etc.). Unfortunately, as far as I'm aware, we have no such textual data for cities of refuge. Thinking through what that evidence would look like leads me to believe (and here I'm just speculating) that we would either need to find administrative records or something along those lines that tracked who was claiming asylum and for what reason. Without records like that, we're unlikely to know.

As for the Hasmonean Period specifically, I'm pretty doubtful. To be sure, the Torah was likely in some kind of near-canonical form by this period and the legal corpora would have been around in some form for a few centuries at least (don't @ me if you're a minimalist). All the same, we don't have much in the way of documentation from that hundred-year period either (and nothing that I'm aware of that attests to the practice).

A few resources that will be helpful for biblical legal material:

David Wright, Inventing God's Law

Kevin Mattison, Rewriting and Revision as Amendment in the Laws of Deuteronomy

William Propp's Anchor Bible commentary on Exodus

Jacob Milgrom's commentary on Leviticus

edit: some formatting and a typo