The quote that you have given here is to the best of my knowledge practically all that is publicly known about this manuscript.
Asides from your quote, the only other place where Christopher refers to this prose version is in his commentary to The Grey Annals in The War of the Jewels. This adds nothing except say that Christopher thinks it was written prior to The Grey Annals (and so even if more detailed, is not the latest prose account of the tale.)
I have described in V.295 how, after The Lord of the Rings was finished, my father began (on the blank verso pages of the
manuscript of AB 2) a prose 'saga' of Beren and Luthien, conceived
on a large scale and closely following the revised Lay of Leithian;
but this went no further than Dairon's betrayal to Thingol of Beren's
presence in Doriath. Unless this work belongs to a time after the
abandonment of the Grey Annals, which seems to me very im-
probable, the two versions of the tale that appear here in the Annals
are the last of the many that my father wrote (for a full account of
the complex history of the QS versions and drafts see V.292 ff.).
Hammond and Scull refer to this text three times in J.R.R. Tolkien: Companion and Guide, but it is unclear if they themselves consulted the original manuscript or not, and so may just be basing themselves on what Christopher already said in those two HoMe quotes.
At about this same time, Tolkien begins to write a prose version of the story of Beren and Lúthien, told at length, but so closely based on the new version of the Lay of Leithian that in places it is almost a prose paraphrase of the verse. He writes it on the verso pages of the ‘later’ Annals of Beleriand, but stops at the point where Dairon tells Thingol of Beren’s presence in Doriath.
J.R.R. Tolkien: Companion and Guide - Chronology - "?Late 1949–1950"
About 1950, Tolkien began to make a revision of the Lay of Leithian left unfinished nearly twenty years before, and a full prose version closely related to the revision. This work included a revision of the story of Gorlim, in which his treachery is less deep and deliberate. Tolkien also told the story of Beren and Lúthien in short in the Grey Annals (c. 1951, see Annals of Beleriand), adding a few details such as descriptions of the refuge of Barahir and his men.
J.R.R. Tolkien: Companion and Guide - Reader's Guide - "Of Beren and Lúthien"
RELATED PROSE VERSION
At the same time that Tolkien returned to the Lay of Leithian, c. 1950, he also began to write another (unpublished) prose ‘saga’ concerning Beren and Lúthien. Christopher Tolkien describes this in The Lost Road and Other Writings as ‘a substantial text, though its story goes no further than the betrayal by Dairon to Thingol of Beren’s presence in Doriath, and it is so closely based on the rewritten form of the [Lay of Leithian] as to read in places almost as a prose paraphrase of the verse’ (p. 295). Written on blank verso pages of the ‘later’ Annals of Beleriand, it follows several earlier attempts by Tolkien to tell the story of Beren and Lúthien in the Quenta Silmarillion at a suitable length. Christopher Tolkien argues in *The War of the Jewels (1994, p. 129) that this new version, which was not known to him when he prepared The Silmarillion for publication, preceded the much shorter versions of the story in the Grey Annals (see Annals of Beleriand).
J.R.R. Tolkien: Companion and Guide - Reader's Guide - "Lay of Leithian"
As for the reason Christopher hasn't chosen to publish it, see "it is so closely based on the rewritten form of the Lay as to read in places almost as a prose paraphrase of the verse". Christopher was very conservative in deciding which texts he included in full in HoMe and which he just referred to in the commentary, (to the point that sometimes his commentary requires opening up other books because he only gives the part of a text that significantly changed from a previous volume). Christopher undoubtedly felt that did due to the similarity between this and the previously published Lay in volume three, that to include this prose text would both eat up pages that could be used for something else and bore the reader with repetition. The stand-alone Beren and Luthien book was intended for an even more general audience than HoMe was, and so the same reasons would apply there as well. (I also do not think it was Christopher's intent to use that book as a venue for publishing new material.)
The manuscript itself is probably in the Bodleian in their "MS Tolkien S" collection. This collection is not available to academic researchers (or for that matter to the general public), without direct written permission from the Tolkien Estate.
The Tolkien Estate has not announced any plans to publish this version