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As far as I recall in real life, the longest scheduled break has been five days. That happened a couple of times in the 1970s when there was a five-day break between the end of the regular season and the start of the LCS.
In terms of punishing a wild card team, there are a couple of things you could do. Have the Wild Card round be a best-of-3 series, but schedule games #2 and #3 as a doubleheader (you'd have to do that manually). Or you could give the advancing wild card team fewer home games in the Division Series round. Instead of a 2-3-2 home field rotation in favour of the division winner, make it, say, 2-2-3 or 3-2-2. Now the division winner gets 5 out of 7 games at home instead of 4. (You'd have to manually make this change.)
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