I need any Brits on the list to weigh in....but I may have spotted a rare instance of John Hillerman's Texas roots bleeding through.
"I fixed pheasant" is indeed an American phrase, though I believe it has spread outwith the south (TM even offers to "fix breakfast").
Over here fixing a pheasant might mean splinting a broken wing (unlikely) or trying to recover from overcooking to the point of dry meat.
However, Higgins frequently slips up. Incorrect intonation, the wrong names or grammatical mistakes, it happens at least once every two episodes.
It could be argued that Higgins has been living over there for so long that he's picked up a few Americanisms but its doubtful given that he holds events for the Queen Mother's birthday, QE2's coronation etc and frequently despises some of those "ghastly American customs".