Books employing the particular terminology that Barth echoes in the cited passage from The End of the Road seem to have flourished between about 1917 and about 1943. A Google Books search for the multiple terms included in Barth's paragraph six such texts from this 27-year period—but the first of these is so clearly presented that you may not need to consult the others.
From "The Lexicographer's Easy Chair," in Literary Digest, volume 55 (October 20, 1917), in answer to an unreproduced question from "J. B.," of Placidville, New York:
When an object is separated by a clause from the governing word the object should be used in the objective case—"Him (not 'he') that is idle, reprove." In interrogative sentences care should be exercised in distinguishing between a pronoun used as the object of a transitive verb or a preposition, and a pronoun used as a predicate complement: as, "Whom (not 'who') did you meet?" "Whom (not 'who') were you talking to?" But "Who (not 'whom') do you think he is?" Therefore, say, "Whom could she have meant?" and "Who could have been meant?"
The subject and the predicate noun or pronoun of a copulative verb, or of a verb in the passive voice, agree in case: as, "I thought it was he." But the noun or pronoun that is the complement of an infinitive derived from a copulative verb is in the objective case, because the subject is in the objective case: as, "We thought it to be him."
The rule that the infinitive or participle of any copulative verb, such as appear, be, seem, etc., may be followed by a noun or pronoun in the nominative case either in the complete subject or in the complete predicate covers the sentences: "I took that (tall) man to be him," and "I thought that (tall) man was he." In English , as in Latin, the verb to be takes the same case after it as it does before it. With regard to the sentence, "He, whom the court favors, is safe," this comes under the rule that an active verb governs the objective case.
Similarly, from John Matthews Manly & Edith Rickert, The Writing of English (1920):
263. After a finite form of the copula or a copulative verb, a pronoun used as predicate complement should be in the nominative case:
It is I—we—he—she—they.
It seemed to be they.
264. After an infinitive with a subject, a pronoun used as predicate complement should be in the objective case:
He took her to be me.
You believed us to be them?
The other four Google Books matches are to texts that, unfortunately, are available only in snippet views:
Kate M. Monro & Sarah Augusta Taintor, Corrective Exercises in English (1928) [e.g., "The predicate complement of a copulative verb is in the nominative case: It is they."]
Joseph M. Thomas, Composition for College Students (1932) [e.g., "G1. Case. G10. General statement. — A substantive is in the nominative case when it is used as the subject of a finite verb, as the predicate complement of a copulative verb, as a vocative, or when it is used in the absolute construction. ... A substantive is in the objective case when it is the object of a finite verb, the subject or predicate complement of an infinitive, or the object of a preposition."]
A. Mortimer Clark & Jaxon Knox, Progress in English, book 2 (1935) [e.g., "When the infinitive of a copulative verb has no subject, the predicate complement refers back to the subject of the principal verb, and is therefore in the nominative case. I should like to be he. ... A pronoun used as the predicate complement after the infinitive of a copulative verb with a subject is in the objective case. He knew the robber to be him. The teacher declared the winner to be me."]
Homer Andrew Watt, Oscar Cargill & William Charvat, New Highways in College Composition (1943) [e.g., "Do not use the objective case when the pronoun is a predicate nominative after the copulative verb "to be." It is he. (Not him.) This is she. (Not her.) It was they. (Not them.) NOTE: "It is me" has been defended as acceptable in spoken English, but not in written English."]
Once you become accustomed to the archaic terminology, the "rule" being asserted seems reasonably straightforward—and at this historical remove from its active dominance, the framing of the rule in terms of "the infinitives of copulative verbs with [or without] expressed subjects" is almost charming.