I believe I have found an, albeit lengthy, workaround. I created a unique ID for each of the paths by concatenating the x midpoint, y midpoint, and z start point of each route (line). I then took the table out of ArcMap and into Excel and reordered the OIDs.
uid(1)
i = None def uid(start): global i try: i = i + 1 except: i = start return i
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