It looks like you people are tying yourselves up in knots over this.
Firstly, by "period", I assume you mean "full stop". I am constantly dumbfounded as to why Americans felt the need to subsitute a perfectly clear word with only one clear meaning ("full stop") for one which shares its meaning with "a quantity of time", "era" and "menstuation".
Enough on that subject, as catamenia is not required in the sentence and a full stop is only required at its end.
To consider the use of the word "how", we might usefully consider the sentence with a substitute which conveys the same meaning:
"In contrast, bronze medalists focus on the fact that if they'd performed slightly worse, they wouldn't have won anything."
One focuses on a "thing". The thing upon which the bronze medalist is focusing on is how he could have lost, in the same way that a prisoner might ask, "how can I escape?" The way that he could have lost was by performing slightly worse.
"if he had performed slightly worse" is not a thing and therefore it cannot be focused upon. Therefore, taking "how" away and leaving nothing in its place would remove the meaning of the sentence.
If anything, "if he had performed slightly worse" could be left out could be left out without turning the sentence into nonsense because it is a secondary clause. To highlight its secondary nature and to clairfy meaning, I would suggest the use of commas as follows:
"In contrast, bronze medalists focus on how, if they'd performed slightly worse, they wouldn't have won anything. "  | 
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