Klein Dictionary gives the etymology of שַׁבָּת as derived from שׁבת (to rest). I also found references suggesting the origin is with the Akkadian šapattum, the 15th of the month, but none related to שׁבע (which originates in Proto-Canaanite).

היּוֹם הַשְּׁבִיעִי (note the definite form!) is how Saturday is referred to in Genesis 2:2-3, without actually having the name שַׁבָּת. There's a some discussion about why שַׁבָּת isn't mentioned there as a name for the day, but regardless, your comment is the first time I read of the etymology of שַׁבָּת deriving from שׁבע.