Gamecock is a Vincenzo Bernardelli model name. Woodcock was another. Stoeger was the importer of V. Bernardellis back in the '50s and '60s, and maybe beyond, I don't remember. None were ever made as a Stoeger. Galef (sp?) used to import Berettas.
I don't believe Bernardelli ever designated a gun "Gamecock". Pretty sure that was a name Stoeger came up with, but not sure when. My Shooter's Bible (1961) still refers to the base model Bernardelli as "San Umberto 1", which later became the Gamecock model. I've never heard of a Bernardelli "Woodcock" model. John Boyd, former Bernardelli importer, would likely know for sure. In that 1961 catalog, the Roma 6 sold for 3x the price of a San Umberto 1 ($450 vs $150).
I'm not a huge fan of that pistol grip either. But with the nice hard case, I'd say it's at least a $2,000 gun. That's not a lot more than the current price for an entry level Uggie. Mark Beasland recently had a new in the box Gamecock 12ga. As I remember, he had it priced around $1400, which I thought was a good deal in today's market.