superlative + relative clause

An earlier question (Relative clauses: “I did the best I could.”) asks about the antecedent of the relative clause, and there are two answers there:

The one (by @Man_From_India) accepted as the best answer by the question owner @Araucaria says that the fused-head NP (the best) is the antecedent, whereas the other answer says “best” itself is a noun, which @Araucaria rejects.

I agree with the fused-head analysis but do not agree that the antecedent of the relative clause is the best. I believe it’s only best, the fused head itself, that is the antecedent, not the entire the best, because the entire nominal best I could is first formed and then the determines that nominal.

That said, whichever answer you choose — between the two answers and mine — all three answers agree that the relative clause (I could) is a postmodifier within an NP (the best I could).

But relative clauses can also be found in an AdjP (Adjective Phrase) or even an AdvP (Adverb Phrase), according to CaGEL* (p 1060):

Here, CaGEL analyzes the phrase (the fastest that he’s ever been) in [8i] as an AdjP, not as a fused-head NP. But I wonder why it can’t be analyzed as a fused-head NP with fastest being the fused head of fastest and person.

Now, CaGEL does agree with the fused-head analysis in other cases (p 1054):

In [65iii], CaGEL treats the phrase (the biggest that I’ve ever seen) as a fused-head NP.

If the answers to the quoted question are right in that the phrase (the best I could) is a fused-head NP, and if it’s correct to analyze that [8i] has an AdjP whereas [65iii] has an fused-head NP, then how do you determine whether a phrase seemingly headed by a superlative adjective is a fused-head NP or an AdjP?

*The Cambridge Grammar of the English Language by Huddleston & Pullum