Very interesting question with, I fear, no simple answer.
I haven't seen the Anatomy of the Ship book yet, but a number of reviewers have had some negative things to say about it. I have the impression (I emphasize again that I haven't seen the book itself) that the author ignored some important sources. I'd better drop the subject of that particular book there, till I've seen it.
To my knowledge the most definitive source on the subject of the Constitution's appearance, and how it's changed over the years, is Capt. (Ret.) Tyrone Martin. He was the ship's commanding officer for a number of years in the 1970s, and his book about her, A Most Fortunate Ship (available now in a second, revised edition) is the best piece of scholarship about her that I know about. Captain Martin, on the basis of a great deal of research in primary written documents and contemporary illustrations, came to the rather surprising conclusion that she didn't have hinged gunport lids at all during her early career. Apparently the ports were closed with some sort of temporarily fixed boards, rather than the hinged ones that I would have taken for granted.
The first time I read that one I found it hard to swallow. But I have a great deal of respect for Captain Martin. I also took a look at a number of contemporary pictures of other American frigates from the period. There aren't many of them, but several of them (most notably those by the French artist Jean Jerome Baugean) are beautifully done and highly detailed. All the Baugean engravings of American sailing warships that I've happened to see have at least one thing in common: they don't show any hint of gunport lids. Neither do most of the contemporary paintings of War of 1812 naval battles.
The Revell 1/96 Constitution is based on a set of plans drawn by George Campbell on commission from the Smithsonian, back in the late fifties or early sixties. Mr. Campbell was one of the best in the business; his plans certainly deserve to be taken seriously. He - quite correctly - used as one of his major sources of information the well-known "Isaac Hull Model" of the Constitution, now in the Peabody-Essex Museum in Salem, Massachusetts. The Smithsonian model has one-piece gunport lids (painted, if I remember correctly, yellow on the outsides and white on the insides). The Revell kit mimics that model - the modern one in the Smithsonian. But the old model in Salem doesn't have gunport lids.
The latter in itself isn't definitive. The level of detail on that model is extremely crude; I can easily believe that the guy who made it wasn't up to the task of making miniature gunport hinges. (I suspect that was Mr. Campbell's logic in adding the one-piece port lids to his plans.) But it does look to me like the preponderance of the evidence favors ports with no hinged lids.
The ports on the maindeck almost certainly had to be sealed with something when the guns were run in - especially in rough weather. But what those removable lids looked like, how they were fastened into place, and what was done with them when the ports were opened I have no idea.
One-piece hinged portlids certainly were in use well before 1812, and I think I've seen contemporary pictures of two-piece ones, with semi-circular cutouts for the gun muzzles, from about that time as well. If I were building a model of the Constitution in her War of 1812 configuration I think I'd be inclined to leave the lids off. But I don't think anybody has an absolutely definitive answer to this one.