First of all, there are very few "British" usages that are not acceptable and used in the U.S. They tend to be in a higher "register," more formal, and more frequent in careful and educated speech.
In my opinion and experience (U.S. native speaker, Northeast, in my sixties) all of the following would be natural and acceptable in U.S. English.
"That's the biggest fish I ever saw."
"That's the biggest fish I've ever seen."
"That's the biggest fish I have ever seen."
"That's the biggest fish I ever have seen."
I think both usages can be found in the U.S. in both formal and informal contexts.
Taking my cue from Susan612, evidence from song lyrics. Hmmm... as I collect examples, it's harder to find examples of the present perfect than I expected.
"The prettiest girl I ever saw
Was sipping cider through a straw."
"Now once upon a time, there was a tree
The prettiest little tree that you ever did see."
"I never saw a moor,
I never saw the sea;
Yet know I how the heather looks,
And what a wave must be."
Will Rogers: "I never met a man I didn’t like."
"I never saw a Purple Cow,
I never hope to see one..."
"I've Never Been In Love Before"--Frank Loesser, 1950s Broadway song title
"I've never listened in among the sounds
That a brook makes in such a wild descent"--Robert Frost
"'If there's anything I do particularly despise, it's a lynching mob; I've never seen one that had a man in it.'"--Mark Twain, and within quoted dialog (a character's words)
"“Yes, it will be lovely,” agreed Hinpoha. “I’ve 1 never lived in such a quiet place before. And I’ve never had you to myself for so long.”--1916 children's book, "The Camp Fire Girls at Onoway House"