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So I am looking to eventually lateral from NY, where I am working at a V10 in litigation, to DC to pursue similar litigation/regulatory work. I've done some searching on firm websites and it seems that DC is very, very COA heavy. For example, Sidley DC's office has 66 COA clerks and 35 district clerks. If I want to work at a top-ten firm in DC, how much do I need a COA clerkship to be in a good position to lateral? Would a firm like Covington care about, e.g., N.D. Ohio at all?

Correlation is not causation. IME, a lot of people who clerk right out of school either worked in DC as an SA or really targeted DC post-clerkship. Firms value clerkships for the experience for a new lawyer, but I don't know that they will value a clerkship as much for a lateral who already has litigation experience. Unless you really just want to clerk, you might want to try lateraling first.