How to tell if something is a core complement or a non-core complement?

In CaGEL's terminology, it's clear as day how to figure out whether a complement is a core or non-core complement.

Ditransitive/monotransitive contrasts

i I gave her the key. I gave the key to her.

ii I envied him his freedom. I envied him for his freedom.

iii They offered us $100. They offered $100

iv They fined us $100. They fined us.

In i–ii the contrast is between a ditransitive construction containing two internal core complements, Oi + Od, and a monotransitive one containing Od + a non-core complement with the form of a PP, while in iii–iv it is between a ditransitive construction and a monotransitive containing just one internal complement (Od).

(Page 297)

Here, CaGEL says both her and the key in I gave her the key are core complements, and that both him and his freedom in I envied him his freedom are core complements, even though you can have a grammatical construction even without her or his freedom as in:

I gave the key.

I envied him.

That is, CaGEL says NP complements that can be omitted, as well as those that cannot, are core complements.

Also, see:

Core complements are generally more sharply differentiated from adjuncts than are non-core complements, and there is some uncertainty, and disagreement among grammarians, as to how much should be subsumed under the function complement.[Footnote 8]

[Footnote 8] Some restrict it to core complements, taking the presence of the preposition in, say, He alluded to her letter as sufficient to make the post-verbal element an adjunct; this makes the boundary between complement and adjunct easier to draw, but in our view it does not draw it in a satisfactory place, as will be apparent from the following discussion.

(Page 219)

Note in the footnote that CaGEL treats to her letter in He alluded to her letter as a non-core complement even though the PP cannot be omitted.

In CaGEL, therefore, whether you can omit a complement is a non-issue in determining whether it's a core or non-core complement. All PP complements are non-core complements; and all non-core complements are PPs.

You seem to have missed (in your examples) that (as stated in the first few paragraphs) a sentence can have two core complements.

What CaGEL implies but doesn't state is that a core complement is one that doesn't use a preposition (we infer its case from its position).

"John bought Mary a cake" has two core complements (dative indirect object 'Mary' and direct object 'a cake') and no obliques.

"John bought a cake for Mary" has one core complement (direct object 'a cake') and one non-core complement (dative shift 'for Mary') with one oblique ('Mary').