Just what is the authentic color for the bottom of a model of a copper-sheathed ship has been a subject of much discussion over the past few years. Personally, I'm inclined to agree with Subfixer. But there are other ways to look at the question.
Different authors have said different things about the appearance of copper after it's been submerged in salt water for a while. Some claim that the motion of the water as the ship moves through it "scours" the copper, and keeps it bright in color; that it only starts to turn green (and various other colors) when it's exposed to air (i.e, when the ship is hauled out of the water). The metal-sheathed hulls I've seen have generally had a mottled, brown and green appearance.
Then there's the question of just what the metal in question was. When we're talking about the Constitution it's pretty safe to assume it was actually copper. As the nineteenth century went on, though, the metallurgists (if they can be called that) made improvements, by adding zinc and other metals to the sheets. That seems to have made the stuff look more yellow (hence the term "yellow metal).
One thing that we can be sure of: though metal sheathing prevented critters from boring holes in the wood planks, it didn't stop things from sticking to the hull. In real life, a hull that had been in the water for a while would have had lots of stuff like seaweed and barnacles sticking to it. I don't think anybody's interested in making 1/96-scale seaweed or barnacles.
Bottom line: to each his/her own. I've seen some beautiful scratchbuilt models with shiny copper sheathing - and some equally beautiful ones with green or otherwise weathered copper. My personal preference is for the latter, but others are perfectly free to disagree.
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