Here a chronology of The Gothams History, as it has been discovered and published by John Thorn and others:
1837: The first organized baseball team is created, called the Gotham Base Ball Club of New York. They played at an open field used parade ground the army called The Parade, at what is now Madison Square. These early clubs were created for exercise purposes, and were restricted to playing catch, shagging flies, and intra-squad games. The best evidence for this original Gotham Club comes from an 1887 interview by the San Francisco Chronicle with William Rufus Wheaton, a lawyer who was a founding member of the Gothams, then went on to become a founding member of the Knickerbocker Club. He served as umpire at what is regarded as the first baseball game recorded in a newspaper on Oct 6, 1845, an intra-club Knickerbocker game. He was also the guy who really first wrote down the rules, not Alexander Cartwright, who got inducted into the Hall of Fame for that, as well as coming up with 9 men, 9 innings, and 90 foot basepaths, none of which he did either. Ref: http://ourgame.mlblogs.com/2012/02/12/how-baseball-began-william-r-wheaton-tells-his-story/
1839: The New York Base Ball Club is created. This is according to a Sporting News interview in 1986 with Harvard trained physician Daniel Lucius "Dock" Adams, the inventor of the shortstop (the ninth position). http://sabr.org/bioproj/person/14ec7492
The pre-existance of the NY Club is backed up by Henry Chadwick, a cricket reporter who became one of the first baseball writers. Look him up, he had a fascinating career.
1840: The Eagle Club is created. They played by rules somewhat different than the other teams (variants were common), which played what has come to be called the New York Game. Some references call the Eagle game "town ball." They officially renamed themselves the Eagle Base Ball Club in 1852 and adopted the Knickerbocker rules. They apparently played by the Harlem River Railroad Depot at 4th Ave (now Park Avenue) and 27th Street, as did members of the Knickerbocker Engine Company No. 12, which included Alexander Cartwright. http://sabr.org/bioproj/person/14ec7492
1843: The Magnolia Base Ball Club is created. They played at the Elysian Fields in Hoboken, as most of the other NY clubs would eventually. http://thornpricks.blogspot.com/2007/11/really-good-find-more-magnolia-blossoms.html
1845: The Knickerbocker club is finally organized. Just as the Knicks are erroneously credited with being the first team, they are credited with playing (and losing, 23-1, to the NY Club) in the first game between teams, on June 19, 1846. There were actually 2 games the previous October, on the 21st and the 25th, between the NY Club and the Brooklyn Base Ball Club, which was made up of players from the Union Star Cricket Club. http://www.la84foundation.org/SportsLibrary/JSH/JSH1980/JSH0703/jsh0703q.pdf
1850: The Washington Base Ball Club of New York is created. http://sabr.org/bioproj/person/14ec7492
1852: The Washington Club is renamed the Gotham Club. This is your team. http://sabr.org/bioproj/person/14ec7492
HOWEVER....as Thorn notes, the Gotham Club appears to have also been referred to as the Washington Club, denoting its primacy (as in G. Washington), the Gotham Club seems to have spawned the NY Club as well as the Knickerbockers, and the players of the Gothams, NY, and Washington seem interchangeable.
In his latest e-mail to me he stated that "The Gothams, aka the New York Base Ball Club and the Washington Base Ball Club, continued to operate after spinning off the Knickerbockers in 1845. There is no record of them in 1847 or 1848 but they reorganized as the Washington BBC in late 1849, then resumed play under their original name of the Gotham BBC in 1852. They endured into the 1870s."....so your Gotham Club might correctly be called the direct continuation of the 1837 Gotham Club....or it may not.
Another problem is the question: what is baseball? Modern baseball has unquestionably grown out of the Knickerbocker Rules, which is essentially the New York Game. But where did the NY Game come from? Al Spalding was a star pitcher for the Boston Red Stockings (today's Braves) for the entire life of the first pro league, the National Association (1871-1875) then jumped to the Chicago White Stockings (today's Cubs), where he helped the owner overthrow the power of the eastern teams, disband the NA and reorganize into the National League. He also started the Spalding sporting goods empire in Chicago, eventually employing the baseball writer Henry Chadwick as editor of his annual Spalding's Official Baseball Guide. Chadwick and Spalding good-naturedly sparred about the game's origins, often publicly....Chadwick claimed it evolved from the English game of rounders, Spalding claimed it evolved from the American game of Old Cat, which had variants. In 1905 Spalding assembled the 7 man Mills Commission, stacked with old baseball and business buddies, which in 1908 produced the Abner Doubleday claim, later proven undisputably as false. In Philadelphia they played Town Ball, with an organized club forming in 1831....if town ball can be considered an early form of baseball, maybe this is the first team. Then there was the Massachusetts Game, locally called simply Ball...and references to base ball, baste ball, base, bace, and others date back to the Revolutionary War, even back to England, where there was "base ball" played, though significantly different from our game. It's still played in Wales. When does a game become "baseball"? In 1823 three NY newspapers, the National Advocate, the New-York Gazette, and the General Advertiser mention a baseball being played in what is now Greenwich Village. Pittsfield, MA, banned baseball in 1791. Even the original Gotham Club may owe its origins to an older club formed in 1832.
Anyway, the amount of information is exhaustive (and exhausing), although there are several gaps. But I think you can and should absolutely amend your team's bio, at least to stop giving the Knickerbocker Club credit they don't deserve, and take more for yourselves. I can give you more, if you like. I am actually trying to compile all this stuff...I started out just trying to find out where all the 19th century teams played. It was a ballpark project. When I discovered the origins of the game, the early days, were shrouded in mystery, I was hooked. Especially when I saw that a lot of the early ball fields are gone, buried under modern Manhattan.