I've got the "Little Office" as well, published by "Carmel Books," dated 1997, but it appears to be a reprint of a 1914 edition.

ISTM that the liturgical construct of the Office is the same, regardless if it's intended for private or public use. There are versicles and responses, psalms with antiphons, responsories, readings, litanies, etc.

What do you mean by "suit it for public worship"?

The little office is laid in the Breviary (EF) and is ancient. King St Louis of France used to pray it. St Charles Borromeo gave it its definitive form. From the papal documents at the beginning of the breviary, it even seems that in old times (before St Pius X, or even before St Piux V), choirs had to join it to the divine office, (matins and lauds together after the day's lauds, the other hours between the finaly prayer and the 'dismissal' except may be at prime and certainly at compline: inserted before the "Converte nos Deus salutaris noster"). This double office was made optional long ago and completely removed by St Pius X.
Still some pious lay people use it in whole or in part (especially those who wear the scapular if not exempted), and may be some nuns in congrations without the obligation of the divine office itself.
Most of the pieces are taken from the divine office and their melodies can be found in the Liber Usualis. The "Antiphonale Romanum" features the few specific ones (e.g. all saints' memory at the end of Lauds and Vespers, brilliant 7th mode, although it might have been suppressed by 1960 rubrics) and the capitules with flex and meter; it also gives some antiphons at Magnificat or Bendictus, which are the same as the Little Office's, in the Saturday Office of Our Lady, but are not found in the Liber Usualis. There is one copy of the "Antiphonale Romanum" at the library of the School of Music at the CUA!

Jam: The Little Office is modeled after the Divine Office, the purpose of which is to praise God. I would think, then, that the Little Office is closer to an akathist.

The second edition of Baronious Press' Little Office has the following comment on its copyright page:

The Nihil Obstat and Imprimatur, given in accordance with Canon Law, are official declarations that a book or pamphlet is free from doctrinal or moral error. No implication is contained therein that authorizes the publication as an "official text for use in the sacred liturgy." (See Canons 822-832, CIC 1983.)

Inter Oecumenici, the first instruction on the Constitution on the Liturgy from the Second Vatican Council has the following to say about little offices:

80. No little office can be classified as conformed to the divine office if it does not consist of psalms, readings, hymns, and prayers or if it has no relationship to the hours of the day and the particular liturgical season.

81. But little offices already lawfully approved suffice for the time being as a sharing in the public prayer of the Church, provided their make-up meets the criteria just stated.

For use as part of the public prayer of the Church, any new little office must have the approval of the Holy See.

82. The translation of the text of a little office into the vernacular for use as the public prayer of the Church must have the approval of the competent, territorial ecclesiastical authority, following approval, that is, confirmation, by the Holy See.

The copyright page for John Rotelle's 1988 little office has the following to say:

This Little Office of the Blessed Virgin Mary is excerpted and adapted from the Liturgy of the Hours.

Various websites that sell this book say it was approved by the CDW, but the book itself makes no claim. A few excerpts from its forward:

With the reform of the Divine Office or Liturgy of the Hours by the Second Vatican Council the guidelines emphasized that this Little Office had to conform to the general thrust of the Liturgy of the Hours so that it would be a prayer of the Church. Taking this into consideration the revision of the Little Office contained in this book follows the liturgical reform of the Divine Office. [...] Those who pray this Little Office join themselves to the prayer of the Church and through Mary unite themselves with that hymn of praise sung throughout all ages in the halls of heaven[...].

It apparently views itself as liturgical; I can take only so much ICEL English (not to mention that the revised Divine Office is a little short as far as the music goes). As far as I know, this only exists in English.

The present Enchiridion indulgentiarum says that the Little Office of the Blessed Virgin Mary is indulgenced, concession 22 (but which version?):

parvum officium legitime adprobatum pie recitaverit (e.g. Passionis D.N.I.C., Sacr.mi Cordis Iesu, B. Mariae V., Immaculatae Conceptionis, S. Ioseph)

I'm not sure if any of these bring us any closer to an answer.

@jdan: The differences are minor and are described on page xvi of my copy of the Baronious Press. Instead of response Dominie, exaudi orationmen meam, it is the Dominus vobiscum and for Matins, instead of Jube, Domine, benedicere, Jube, domne, benedicere.

@jgirod: When I have too much time on my hands, I double some of the hours.

Which version? Any "legitimately approved" (legitime adprobatum) version would suffice: if the list of examples is an indication, apparently there are "Little Offices" designed around other themes besides those of Our Lady.

Some "Little Offices" appearing on the net, presumably from old books, seem very truncated, lacking psalms, canticles, and a scripture reading, so they would not comply with the point under n. 80 above.

E.g.: "The Little Office of the Sacred Heart (perhaps only parts of it?)
http://www.preces-latinae.org/thesaurus/Filius/OPSCI.html

...of the Immaculate Conception
http://www.catholic.org/prayers/prayer.php?p=1925