DrJambo
Crash-test dummy
Everyone should take a read at Catt's post. For me that sums up culture flipping and details the best ways to prevent such annoyances occurring. 
As mentioned by Zach, the way to have dealt with it, if you'd been lucky enough to get a few more turns would have been to mash the surrounding cities to prevent the pressure form the tiles you don't control being such a heavy factor (10 tiles!). One of the reasons i think culture flipping was introduced was to prevent just this.... attacking deep within the heart of enemy territory to claim enemy cities. Most wars are fought with the external borders advancing/retreating, very rarely are countries taken from the inside first!
Even if you had razed it and used your own settler, i don't think it would have made too much difference. The pressure from the close proximity of American's Capital and the tiles you don't control would still have been there.
Much better would have been to attack one of the American's peninsular cities further away from their Capital and nearer yours and your own borders....or just to have razed the *******

As mentioned by Zach, the way to have dealt with it, if you'd been lucky enough to get a few more turns would have been to mash the surrounding cities to prevent the pressure form the tiles you don't control being such a heavy factor (10 tiles!). One of the reasons i think culture flipping was introduced was to prevent just this.... attacking deep within the heart of enemy territory to claim enemy cities. Most wars are fought with the external borders advancing/retreating, very rarely are countries taken from the inside first!
Even if you had razed it and used your own settler, i don't think it would have made too much difference. The pressure from the close proximity of American's Capital and the tiles you don't control would still have been there.

Much better would have been to attack one of the American's peninsular cities further away from their Capital and nearer yours and your own borders....or just to have razed the *******
