While the BBC might spell it that way (and not consistently...), most scientific publications and museums seem to refer to Smilodon is one of the sabre-toothed cats.We frequently say 'fine tooth comb' rather than 'toothed', just as Smilodon is a Sabre Tooth Tiger.
I really don't know why the original thread went wandering off to discuss a variant of the fine-tooth comb that had not been mentioned - except to point out that it might be considered wrong.Another idiomatic expression, at least in American English, would be:
"I got stuck on the board for two hours. The customs officers went through my car and my bags with a fine-tooth comb."
I thought I might point out that the OED does recognize 'fine-toothed file' and 'fine-toothed saw' so the structural logic behind 'fine-toothed comb' isn't in question.BE speakers have been using fine-tooth combs, as JamesM wrote, for a long time. It is not a mis-spelling. It is the original and only form admitted by the OED. The verb is to fine-tooth-comb.