Fascinating subject, worthy of much more research. The "Berber Quran" did not survive, though bits of it are reported by others, if you can believe them, which is a question mark. This is a religious movement in a specific Berber tribe, the Berg̲h̲awāṭa/Bargawata (you'll find different spellings in the literature, makes search a bit harder) There's more than one source here, and it seems fair to say that "yes, this really did happen-- and yes, what we're hearing is coming from unsympathetic historians", so my read is that your skepticism is well founded.

Says Rene Basset:

Il n'y a donc aucune source ecrite par un Bargawatien ou par un sympathisant de cette religion, car toutes les informations qui nous sont parvenues emanent de leurs contradicteurs

trans:

"There is no source written by a Bargawatien or by someone friendly to this religion, as all the information that comes down to us derives from their critics

So far as I'm aware, we rely mostly on Ibn Khaldun, the geographer Ibn Hawqal, Abū ʿUbayd al-Bakrī and a little known writer Abu Ga'far Ahmed b. Nasr al-Dawudi ("al-Dawudi") for what we know of this. None are friendly to the Berg̲h̲awāṭa. Salih ibn Tarif and his father Ṭarīf Abū Ṣāliḥ were both supposed to have been Kharijites / (khawarij) -- a term that means, sort of "the leavers"; people who broke with the mainstream of Islamic opinion over mostly political issues ("the first fitna"); unpopular with the local Malikites and Sunnis generally. The Bargawata, inspired by the Kharijites, were the nucleus of the "Berber Revolt" (AD 729, AH 122), and I think it's a fair inference that Salih ibn Tarif is part of this tradition of rebellion.

There are details of Salih ibn Tarif's "Quran" reported, suggesting that it followed the Quran very closely in format. So far as I can determine, the Chapter titles are known, -- or at least claimed to be known-- mostly names of prophets cited in the Quran, from Ayyub to Yunus, but no copy exists.

Here's a bit of what we're told of their religious practices:

In repeating the profession of faith they hold their hands open with the palms downwards. At public prayer, which was celebrated on the Thursday and not the Friday, they repeated half of their Ḳorʾān, standing and the remaining part, while making their prostrations. The ceremony was completed by repeating in Berber the following formulas: God is above us, nothing which is on the earth nor in the heavens is hidden from him. Then they repeated twenty times the formula: Moḳḳor Yaḳūs̲h̲ = God is great; Ihān (Iwen?) Yaḳūs̲h̲ =: God is one; ur d am Yaḳūs̲h̲ = these is none like God. It is evident therefore that with the exception of the use of Berber their religion did not differ essentially from Islām.

. . . and it's Ibn 'Idari l-Marrakusi who tells us of "five prayers a day and five at night, albeit some without prostration (sujud).

The Berber Kingdom survived for a time, but was ultimately crushed by the Almohad, Abd al-Muʾmin.

As one parallel, you might consider the Druze. They're unrelated, but this is a case where we can more clearly examine a "heresy" in Islam-- how Muslims portrayed it, and what it objectively looks like. Some of the more dramatic claims made by contemporary Muslim historians about the Druse were true-- that is, for example, they really do drink alcohol, and some are exaggerated. In the same vein, when I read the allegations about unusual marriage/divorce practices among the Bargawata, I wonder whether that's not a reflection of some real Berber traditions-- the Berbers were (and remain) known for having different traditions to Arabs in the relationships between men and women, something that I could imagine being played up by a hostile historian.

Sources:

The most extensive discussion I'm aware of is from 1910:

Recherches sur la religion des Berbères (Paris, 1910), p. 48—91. It's in French, though it indicates an English translation in the Encyclopaedia of Religion and Ethics edited by James Hastings (which is online, at least partially).

Encyclopedia of Islam, 1st edition - see the entry for Berg̲h̲awāṭa (the Encyclopedia of Islam is an invaluable resource, but it's behind a paywall; you can likely get access if you're at a university library).

Texte méconnu sur deux groupes hérétiques du Maghreb médiéval -- a lot of the material on North Africa is in French. This is a recent article (2005) by Allaoua Amara, the leading authority in the field and if you can read French, its where I'd start.

Kitab al-Amwal - this is al Dawudi's only surviving work, where he has his most extensive discussion of this subject. As a Malikite, he was not at all sympathetic to al-Khawarij or to the Bargawata.

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