I would have to say that the Bowie is THE American knife, but the problem that you have is which Bowie? There are the "James Black*" designs from Washington, Arkansas, and those based upon them, which seem to all have the sharpened clip, and then there are the J.D. Searle versions from Baton Rouge, Louisiana. These are the ones that fairly closely approximate the Spanish fighting knives and were ones which Rezin Bowie was accustomed to giving to friends, since his plantation was near Baton Rouge. There is a sheath for one of these in the Alamo Museum, I have read. Dixie Gun Works sells an excellent replica of the J.D. Searle Bowies.
As to the James Black Bowie design, this is the one that is our image of the "classic" Bowie for most of us. It is a large knife with a clipped point blade that has a sharpened clip and a wicked point capable of mean back-cuts (as taught by James Keating) and frequently has some sort of file work on the spine to momentarily catch an opponent's blade, as in the case of the Bart Moore Bowie. I would have to say that the best modern interpretations of the classic Bowie Knife are those by Bill Bagwell and their production versions by Ontario, remembering that the Bowie was, at heart, a fighting knife, and that is what the Bagwell Bowies are, uncompromised fighting knives.
But may I also suggest one other possible "American" knife, the Barlow clasp knife. I don't believe that I have ever seen one from a European maker, but it has an American tradition going back to the early 19th Century, at least. It is still made and is one of my favorite designs. It was the first knife that I ever owned, actually, long since lost. I have since bought a couple of high quality examples, just to keep around for "auld lang syne."
* I once caught unmitigated flack in this forum from the followers of Bernard Levine when I was a newbie for suggesting the "James Black Version" of the Bowie Knife's history. It seems that he disagrees with it and has said so quite often and quite loudly. He is quite expert and his opinions are not to be taken lightly in these matters, but he is only one of many experts and I have read others who disagree with him just as often and just as loudly.
I also would observe that, over in SwordForums, there have been some discussions of Seax Knives that have pointed out how the design of these knives varied widely over the extesive time ansd geographic periods of their use and that there were seaxes that did look remarkably like Bowie Knives, especially, apparently, among their users in the Eastern Route through Russia.