In what ways do Christian denominations reconcile the discrepancy between Hebrews 9:27 and its Biblical counter-examples?

OP: Given that there are counter examples to the notion that men only die once, how do denominations apply a verse at times literally and at times figuratively?

To answer, typically the context will show the intent, like a parable.

The examples the OP provided about Heb 9:27 are literal, but do not prove the point trying to be made. Both deaths occurred prior to Christ's death, burial, and resurrection. So, death, then judgment to come would still be a future event for those in Christ and apart from Christ.

Clearly Paul knew that some people do die more than once, having resurrected someone himself:

As Paul spoke on and on, a young man named Eutychus, sitting on the windowsill, became very drowsy. Finally, he fell sound asleep and dropped three stories to his death below. Paul went down, bent over him, and took him into his arms. “Don’t worry,” he said, “he’s alive!”
…
Paul continued talking to them until dawn, and then he left. Meanwhile, the young man was taken home unhurt, and everyone was greatly relieved.
— Acts 20:9–12 (NLT)

The key to understanding Hebrews 9:27 is in the words translated here as "appointed" and "once".

Strong's Greek Lexicon — G606 says that "apokeimai" means:

• to be laid away, laid by, reserved
• reserved for one, awaiting him

The word refers to something in one's future.

Similarly, the meaning of the word translated as "once" doesn't have to be a specific restrictive count. It could mean at least once, or even at some time or inevitably.

For instance, in Hebrews 6:4, "those who were once enlightened" isn't claiming that these people were restricted to exactly one enlightenment in their life.
And more significantly, in Hebrews 12:27, the same Greek word for "once" is translated as "once more", clearly indicating that "once" doesn't necessarily refer to the only instance of something.

Hebrews 9:27 is simply pointing out that death awaits everyone, not explicitly setting a restriction.

Even though it is a common sense understanding that, by and large, people only die one physical death, the main point of the passage in Hebrews 9 is not to delineate how many times a person may die. The main point is that following death there is judgement rather than something else, like a second chance, and Christ came according to that pattern:

For Christ is not entered into the holy places made with hands, which are the figures of the true; but into heaven itself, now to appear in the presence of God for us: Nor yet that he should offer himself often, as the high priest entereth into the holy place every year with blood of others; For then must he often have suffered since the foundation of the world: but now once in the end of the world hath he appeared to put away sin by the sacrifice of himself. And as it is appointed unto men once to die, but after this the judgment: So Christ was once offered to bear the sins of many; and unto them that look for him shall he appear the second time without sin unto salvation. - Hebrews 9:24-28

As it is appointed unto men once to die, but after this judgement: So Christ was once offered to bear the sins of many.

It is not wrong to understand that, generally speaking, everyone dies one physical death each. It is also not wrong to understand that God is able to raise the dead both in a permanent sense in Christ and in a temporary sense for some specific purpose, like testifying to Christ (as in Lazarus leaving the tomb at His call).

It is this general tendency for everyone to die once that the writer of Hebrews uses to point out both the temporary, inadequate nature of the Levitical system (even those priests sin and die) and the eternal and perfect priesthood of Jesus who, without sin, died, not for his own sin but for ours.