Why did Herman Melville write "if he have a chance" in Moby Dick?

if he have a chance he will pull down and pulverize that subaltern's tower

The OP wonders why we see the plain form of the verb have and not present tense has in the excerpt above. The reason is that the clause shown above is a conditional construction, and the smaller if-clause, if he have a chance, is a conditional protasis (the 'protasis' is just the bit that contains the subordinate clause of the conditional).

In older Englishes, a conditional protasis was one of the special environments which licenced the use of subjunctive constructions. Subjunctive constructions are a special type of finite clause. They use nominative subjects (when the subject is a pronoun), but have a plain, untensed form of the verb.

In modern, Present-Day English, we most often see subjunctive clauses when they occur as the complements of verbs indicating some type of command, wish or preference:

1. She insisted that he stay with her.

In the sentence above, we see the plain form of the verb stay instead of either stayed or stays. All three options would be possible in Present Day English (henceforth PDE), but the subjunctive is currently decidedly more frequent in American English than in British (Waller 2017).

In PDE, subjunctive clauses are still frequently found in conditional protases. However, the type of conditional protasis and the word introducing it have quite a marked effect on how restricted the use of subjunctive constructions is, and what verbs may occur within them.

Conditional protases introduced by prepositions (read 'subordinating conjunctions') such as provided or unless licence the use of subjunctives:

2. She agreed to embark upon the journey provided he stay with her.
3. She refused to remain on board unless he stay with her.

Although such subjunctive protases are slightly formal or literary, and their use is somewhat restricted, we nonetheless see them occurring with a range of verbs.

There exists a special type of conditional, the exhaustive conditional which contrasts semantically with the if-conditional. In these conditionals, the conditional protasis tells us that it does not matter whether some conditon is met or not, or how it is met; the outcome will be the same:

4. Whether you go there or not, I'm leaving.
5. Whoever wins, I'm leaving.

In exhaustive conditional protases in PDE, we again see subjunctive constructions used with a range of verbs. In such conditionals, the use of the subjunctive is arguably freer and more frequent, and when the verb concerned is the verb BE, the use of the subjunctive is also considerably less stuffy:

6. I'm always up for a hot breakfast, whether that be beans on toast or a full English!

In PDE, out of all the many different types of conditional constuction, the use of the subjunctive is arguably most restricted in if-conditionals. Nowadays, the use of the subjunctive in if-protases is restricted almost entirely to the verb be. Here is an example from COCA:

7. [I]f he be occupied with something unlawful, or even with something lawful, but without due care, he does not escape being guilty of murder.

Notice that in an if-conditional the use of the subjunctive, even with BE, is usually decidedly formal or literary. The following would be considered very affected, perhaps even ungrammatical:

8. ?I'm always up for a hot breakfast if it be beans on toast.

With other verbs, the use of the subjunctive in if-protases is extremely rare, and as we can see from the striking oddness of Melville's sentence to the PDE ear, would most often be considered ungrammatical:

9. *If he have a chance he will pull down and pulverize that subaltern's tower. ^{[Ungrammatical in PDE]}

Even in Melville's day, such use of the subjunctive would be quite literary, and was slowly dying out. That may have made it a more evocative form to use here.

^Reading:

^{For more information on the subjunctive in PDE, see The Cambridge Grammar of the English Language (Huddleston and Pullum, 2002), A Comprehensive Grammar of the English Language (Quirk et al, 1985) or Tim Waller's PhD thesis from the department of English usage at UCL The Subjunctive in Present-Day English. The latter is chock-full of lovely examples and has a lot of interesting information about the frequency of the subjunctive in British and American English in different environments over the last century.}

Mr. Melville is using a form of the subjunctive mood that was still common in formal or elevated English in the 18th and early 19th centuries. In older and more literary English, the subjunctive of have was have, not has.

So: "if he have a chance" = subjunctive (expressing possibility or condition, not certainty) "if he has a chance" = indicative (stating a simple factual condition)

At the time Mr. Melville was writing, the subjunctive was associated with formal, rhetorical, or archaic style, especially in moral or philosophical statements. The construction would have sounded dignified, slightly old-fashioned, and scriptural to Mr. Melville’s readers—much like language from the King James Bible (e.g., "if he have ears to hear").

So in the sentence: “if he have a chance” Mr. Melville is not just stating a condition. He is presenting a general truth about human nature. The subjunctive emphasizes that this scenario is not just one case, but a recurring pattern, almost a law of pride and resentment in human nature.

In modern English, the meaning is still clear, but the tone changes. “if he has a chance” sounds practical and factual. “if he have a chance” sounds universal, moral, and slightly archaic—which is exactly the tone Mr. Melville wanted.