Using commas correctly around a relative clause (bleeding into “that” vs. “which”)

According to the Webster’s New World College Dictionary, 4th Ed., a clause is “a group of words containing a subject and a finite verb.” Relative clauses provide additional information about a noun without the need for creating a separate sentence. The way I remember the correct use of commas around a relative clause is by following the rule “either one uses two commas to get to the predicate of the sentence, or one uses none.” When a clause adds information which is useful but not necessary to the intended meaning of the sentence (a nonrestrictive clause), insert a comma both before and after the clause; when the clause provides information necessary to the intended meaning or context of the sentence, use no commas. Example: “The goldfish I bought last week has grown a full inch.” The clause “I bought last week” restricts the world of goldfish to the one I bought last week, as opposed to:  “The goldfish, which I bought last week, has grown a full inch.” Here, the information that I bought the goldfish last week is unnecessary to my main point that it — and the verb context provides there is only one —  has grown an inch. In the first sentence, I could have written “The goldfish that I bought last week …” AP Style tells us we can and ought to leave out the connective pronoun “that” where it is unnecessary to provide context, and as I prefer tight sentences I tend to apply that rule. However, the Chicago Manual of Style states (in 5.202): “In polished American prose, “that” is used restrictively to narrow a category or identify a particular item being talked about; “which” is used nonrestrictively – not to narrow a class or identify a particular item but to add something about an item already identified.”