This is a good question. Are beers brewed in Belgian Trappist monasteries Belgian or Trappist first of all? Since "Trappist" beers currently proliferated truly originate from recipes developed by monks in the monasteries and sold publicly for centuries to provide for their needs and assist in their intentions/charities and which established much of the backbone of secular-produced Belgian beer styles, that established tradition of monastery-conceived brews suggests that Trappist ales wherever brewed will continue those styles, as passed down for generations.
However, brewing tradition starts with available raw materials plus innovation. Therefore, due to different water profile, and malts, and hops available at good price locally (continentally?) in America, might this spur development of a new style or styles of beer to newly gain the "Trappist" brand?
I concur that there is enough interest in Trappist beers in America, which can be brewed anywhere in the modern age, that the good monks will bank on the reputation and artistry of their forebears, and continue in those same styles, but I would not be surprised by the American adventure as a potential stepping stone for new innovation that could eventually further offerings under the Trappist famous brand.