Is it acceptable to use "is become" instead of "has become"?

The use of "is become" here relates to verbs of motion/transition; verbs of motion would take be while other verbs would take have. There is no such grammatical distinction in English perfect forms anymore.

English began with this distinction, as did sibling languages like German (as Cindi pointed out originally in a now-deleted answer).

Here is what happened after that. This is an excerpt from the OED's discussion of auxiliary have:

In early ME., [have extended its use to the verb to be, as in "have been", like French]. Verbs of motion and position long retained the earlier use of the auxiliary be; and "he is gone" is still used to express resulting state, while "he has gone" expresses action.

This is talking about English retaining the auxiliary be for motion verbs, like present-day German. Originally, the verb "to be" also used be as an auxiliary for the perfect, e.g. "it is been cold", but changed to have in early Middle English. (German still uses "ist gewesen", or "is been", today.) After this change, the other motion verbs still retained the be-auxiliary for perfect.

In Modern English, the motion distinction completely faded out, and be was replaced with have across the board, except in a very specific case. The OED describes this case:

in intr. vbs., forming perfect tenses, in which use it is now largely displaced by have after the pattern of transitive verbs: be being retained only with come, go, rise, set, fall, arrive, depart, grow, and the like, when we express the condition or state now attained, rather than the action of reaching it, as ‘the sun is set,’ ‘our guests are gone,’ ‘Babylon is fallen,’ ‘the children are all grown up.’

Keep in mind that become is not intransitive, so "is become" doesn't work anymore, with any meaning, in present-day English (— except, of course, in poetic use).

The use of 'be' rather than 'have' to form the perfect of some intransitive verbs ("I am come", "I am become" etc) is archaic in Modern English, and used only for special effects.

"Is become" is archaic. The "to be" and "to have" verbs used to follow the model of French verbs in the present perfect (passe compose in French) and the French still follow it. The etre (to be) and avoir (to have) are still used this way and for verbs such as "to come" and "to become", etre would be used in the perfect.

I have come to this site because I just now used a "be" verb with "become", and I wondered how my usage fits with theory and practice as currently understood.

My sentence (put in an electronic letter/note):

(1) No wonder my missives are become so long.

As Kosmonaut prescribes, my usage is quasi-poetic (evidenced by my use of the archaic "missive" to describe my notes). But there may be more going on here than that:

As Heckschei observes, "become" is not necessarily a transitive verb. It is transitive in (2), intransitive in (3):

(2) I wondered how it would feel to be my brother for a day, so yesterday I put on his clothes and went to his job, and in effect I became him.

(3) The window opened, and I became cold.

That (2) is a transitive usage is attested by the objective case of the pronoun; we cannot say (2'):

(2') *I wondered how it would feel to be my brother for a day, so yesterday I put on his clothes and went to his job, and in effect I became he.

But there is clearly no object of "became" in (3), and so that must be an intransitive use.

So we ought to include "become" in the list of intransitive verbs indicating a transition of state that are (or at least might be) eligible for "be" usage in the perfective. But I'm not able just now to come up with any convincing instance of such a usage, i.e., one that doesn't feel archaic. The closest I can come is this:

(4) ?It is become common to use "have" with nearly all verbs in the perfective.

But I'm not convinced (4) is any less archaic-sounding than (1), and it is without the use of "missive" to justify a quasi-poetic usage.

So if "become" is now purely in the archaic/poetic usage for "be" in the perfective, why? There is no more proto-typical change-of-state intransitive verb than "become"; why should that not fit at the head of the class along with "go", "come", "grow", and so on?

Indeed, "come" usage with "be" is rather archaic; the only usage I can think of is Tolkien's:

(5) Ai! A Balrog is come!

That sounds right to my ear, but only because it's right for the elf Legolas to be speaking in archaic mode. It seems odd to me that "go" retains the "be" usage but "come" does not. Does anyone have a counterexample for "come"? If not, maybe there is a link here between "come" and "become". (And maybe "become" originally started precisely as "be" + "come"?)