"Being [he/him] is not easy." Which is prescriptively "correct"?

I cannot offer a systematic treatment of this subject, but I have found one interesting data point, English Grammar Simplified, Its Study Made Easy by James C Fernald, LHD (1916, which, alas, predates your source by three years only). To show that this is clearly a text from the prescriptive school, I quote Dr Fernald's preface:

The facts of correct English usage are, for the most part, as sure as the facts of the working of a watch or of a locomotive. We can do no better service for any student, young or old, than to tell him definitely what those facts are, and let him learn them once for all.

So there.

In the section of pronouns Rule 1 (p.199) for the nominative case makes that case required in predicate nominatives:

1. The Predicate Nominative; as. This is he; It is /.

Caution. — Such forms as, "It's me/' 'That's him," etc., are erroneous.

EGS prescribes the objective case for objects of infinitives and participles in Rule 9 (p. 202) for the objective case:

9. The Object of an Infinitive or Participle; as, My father promised to send them; I have not known of his meeting her.

However, Rule 2 (p. 200) for the nominative case allows for an exception under warning and caution:

2. The Nominative after an Infinitive or Participle; as, I could wish to be he; I did not think of its being /.

This use; however, is rare, and often harsh or clumsy, and therefore in general to be avoided.

Caution. - After an infinitive with a subject in the objective case, or after a participle agreeing with a noun or pronoun in the objective case, the pronoun following such infinitive or participle must be in the objective case; as, I understood it to be him.

(Everywhere emphasis is mine)

The book Higher Lessons in English: A work on English Grammar and Composition, in Which the Science of Language is Made Tributary to the Art of Expression, by Reed and Kellogg, published in 1878, goes into great detail on the grammar of whether you should use the nominative or objective case with forms of the verb to be.

The primary rule for which case to use, which works most of the time, is:

• The cases on either side of the verb to be should match.

I give the relevant quote from the book

A noun or a pronoun used as an attribute complement or a participle or an infinite is in the same case (Nom. or Obj.) as the word to which it relates as attribute.

Examples.—Being an artist, he appreciated it. I proved it to be him.

There are some further rules that cover situations which the primary rule doesn't handle.

1. "When the assumed subject of the participle or infinitive is a possessive, its attribute complement is said to be in the nominative case.

The example they give is:

Its being he should make no difference.

2. "When the participle or infinitive is used abstractly, without an assumed subject, its attribute complement is also said to be in the nominative case."

Example they give for this are

To be he is to be a scholar. Being a scholar is not being an idler.

3. "The assumed subject of the infinitive being omitted when it is the same of that of the principle subject, him, in the sentence I wish to be him—equaling I wish (me or myself) to be him—is the proper form, being in the same case as me."

That is, in sentences like

You can be whomever you wish to be,

there is an assumed object of the verb wish,

You can be whomever you wish (yourself) to be,

and this object is in the objective case, so whomever should also be in the objective case.

Using these rules, it appears that the three examples in case (3) of the question above should all be in the objective case:

• It was thought to be him to whom the speaker referred,

because the infinitive to be equates him and (to) whom the speaker referred.

• I should like to be him,

because there's an implicit pronoun in the objective case in this sentence: I should like (myself) to be him.

• Do you think that you should like to be him?

Again, there's an implicit pronoun: Do you think you should like (yourself) to be him.

For the "gray area" in the OP's question, where there is no antecedent of any kind, the right case according to this book would be nominative because of the second rule above:

• Being he is not easy.
• To be he is not easy.
• It is not easy to be he.

These last examples are nearly the same as the book's example:

To be he is to be a scholar.

"Being he" and "To be he", despite their awkwardness, do seem to have been traditionally prescribed by at least some grammar-guide authors in sentences where there is no preceding subject of any kind. (However, Peter Shor's answer indicates that this was not universal; some authors apparently thought the accusative would be better in some contexts.)

New Graded Method in English Grammar, by Marion Durand Mugan (1890), says

Rule 5. A noun or pronoun used independently is in the nominative absolute case (and should have the nominative form).

Explanation.—The noun or pronoun which is said to be used independently with a participle, is what may be considered the subject of the participle or the predicate following a participle from the verb, to be.

Analyze and correct the following sentences, using this as a model:

• "I had no idea of its being him."

Him is a pronoun used independently with a participle, therefore it is in the nominative absolute case, R. 5, and should have the nominative form, he.

If the argument for nominative case here is based on this being analyzed as a "nominative absolute" construction, then I think the same prescription would presumably apply to pronouns that are the predicative complement of "to be" or "being" without any overt subject.

Nobody actually seems to use the construction with subjectless "being"/"to be" + nominative complement in practice, though, as can be seen by comparing the Google Books results for "easy being me" with those for "easy being I".