Can't decide between She/her

Solution 1:

Traditional/prescriptivist "grammar rules" (the kind that you might be tested on) aren't a complete system for creating natural-sounding sentences

You didn't mention the source of these sentences in your original post. In a comment, you clarified that they are adapted from the following questions in Paragraphs and Essays with Integrated Readings, 12th Ed., by Lee Brandon and Kelly Brandon:

"19. Was it (I, me) you hoped to find?"
"7. Only two were chosen, Kathy and (he, him)."

(p. 512)

This is a typical context where you might be expected to apply prescriptivist "grammar rules". According to that framework, the answers to 19. and 7. would be "I" and "he" respectively.

Note that you aren't being told to write your own sentences, but to choose one option in a sentence that you are given.

The "rules" you use to answer questions like this aren't the rules that English speakers unconsciously rely on to construct natural-sounding sentences, and they aren't even exactly the same as the rules that English speakers consciously rely on to construct formal, but non-archaic-sounding written texts.

The traditional rules don't constitute a fully productive grammatical system. Rather, they explain the form of certain special constructions that are used and thought of as "correct" in rather formulaic expressions. For example, "The only people there were John and I" is much more likely as an example of a "predicate nominative" than the "The only people there were the pastor and we". Writing guides may recognize exceptions to the traditional rules when they produce results that are judged as being too stilted; for example, this post from the Grammarphobia blog, by Patricia T. O’Conner and Stewart Kellerman, says "In all but the most formal writing, “It’s me” is now acceptable" (How should you answer the phone?).

Natural-sounding sentences don't follow traditional "rules" (they do follow rules, just not the same ones)

For me, the natural spoken forms equivalent to the three sentences that you mention would be as follows:

Only one of the contestants was chosen: her.
Was she the one you hoped to find?
Congratulations on beating everyone else in the Pokemon League ... but there's still one more opponent left—me!

Basically for the reasons given in Araucaria's answer. The function of the English pronoun form often called "accusative" is really not analogous to the function of the accusative case in languages like Latin, Russian or German, so you can't just rely on principles like "subjects take the nominative case" and "appositives take the same case as their antecedents" and expect them to apply unproblematically (even in formal English). In a number of contexts, English "accusative" forms have what could be informally labelled a "disjunctive" role: they seem to be used as a kind of default form in a variety of contexts (this is covered in many other posts on this site; e.g. "Who wants ice-cream?" — Should I say "(not) I" or "(not) me"?) From the formal perspective of the modern science of linguistics, Araucaria's formulation relating the "nominative" case in English to finite/tensed verbs seems to be common, and from some perspectives, that makes the English case distinction not really a true nominative/accusative distinction at all. Omer Preminger says "insofar as English has anything you’d want to call ‘nominative’, it's [...] the thing we’ve been calling ‘accusative’ or ‘objective’ case" ("Case in 2017: some thoughts", p. 29).

To sound natural in formal language, rephrase

If I was trying to sound formal, and also to avoid violating any rule that has ever been proposed by any prescriptivist, I would say something like the following:

Only one of the contestants was chosen: it was she.
Was it she whom you hoped to find?
I congratulate you for defeating everyone else in the Pokemon League ... but you still have one more opponent left—me!

Rewording sounds much better than using bare "she" and "I" at the ends of sentences 1) and 3). Replacing "Congratulations on beating" with "I congratulate you for defeating" isn't a matter of grammar, but it increases the formality. Even with heightened formality, I couldn't find a way to make "there is still one more opponent left—I" sound natural, so I would reword instead.

If you can't rephrase ... you can be "correct", but it will sound bad

If you somehow need to stick with the original wording, and just choose which out of "she" or "her" would be preferred by an extreme "stickler" prescriptivist, your reasoning is correct for that purpose. In traditional grammar, "nominative" forms are used to match the case of a nominative antecedent, or as a "default" case in certain context (see some of the citations in my question "Being [he/him] is not easy." Which is prescriptively "correct"? or in the answers there).

Only one of the contestants was chosen, she.

Despite sounding awkward, or possibly outright unacceptable, to a modern English speaker, "she" would be correct according to traditional rules because of the principle of case-matching (with the nominative NP "(only) one of the contestants", as you mentioned), or failing that, because of the principle that nouns take nominative case as independent elements of a sentence.
Was it she you hoped to find?

This certainly ought to be nominative according to the traditional rule of matching the case of the subject and the case of the complement of a copular verb.
Congratulations on beating everyone else in the Pokemon League ... but there's still one more opponent left—I!"

Rather than calling "I" a shortening of ""It is I" in 3), I would say that it is appositive to "one more opponent (left)". But the exact terminology and analysis of "appositives" is fairly messy anyways, and it doesn't lead to a different answer (because according to traditional prescriptivists, "one more opponent" in "There's still one more opponent left" would be analyzed as a nominative NP).

As I said, being able to choose the "correct" option in contexts like this is of limited practical value. You can construct all kinds of sentences that are "technically correct" according to the standards of archaic prescriptivism, but that sound terrible and won't be effective at producing whatever effect you actually want to produce in your readers/listeners.

Solution 2:

You cannot reason your way into making grammar fit the rules you invent for it. The rules of grammar are what actually happens in the language!

The rule is that when a pronoun isn't the direct subject of a tensed verb, it does not need to be in nominative case—and normally can't be.

It doesn't matter at all whether the pronoun represents a Subject or not. If it isn't the immediate subject of a tensed verb, it won't be nominative.

For this reason we see the following data:

A: Who's that?

B: Who? *She? (Ungrammatical)

B: Who? Her? (Grammatical)

In the Original Poster's example, the pronoun she is part of a right dislocation. Here it echoes and identifies the subject one of the contestants. However, this pronoun is not the actual subject of a following tensed verb. For this reason it will not be nominative, but accusative. The accusative is the normal case for pronouns that aren't subjects of tensed verbs.