First of all, I agree on both points made by @MauroAllegranza (as I often do).
Regarding "Anti-Garve" and its origins
Timmermann in the introduction to his commentary on the Groundwork (2007, Cambridge UP) refers to a letter from Hamann to Scheffner (February 1784), where he describes the project as a counter critique ("Antikritik") against Garve's translation of Cicero's De officiis from 1783 and - more importantly - an indirect answer against the (complete) review of Garve as published in the Allgemeine Deutsche Bibliothek.
It should be emphasised that this is from hearsay (the letter begins with "Einer Sage nach", which can be translated as "according to accounts received")!
I could not find a complete English translation of this letter, the German original can be found e.g. here.
The word of the "ever prolific" (xxvii) Hamann is not to be taken too seriously, though, as Timmermann himself acknowledges in his remark:
It was again Hamann who, in a letter to Scheffner in February 1784, reported that Kant was working on a ‘Counter-Critique’ (Antikritik) of Garve’s ‘Cicero’ that was, as a matter of fact, intended as a retort against the unabridged review of the Critique (IV 626). It is difﬁcult to say whether Hamann’s testimony is credible.
In a letter to Herder two weeks earlier, he also says that "it is said that Kant works on" the Antikritik (see above link).
Interestingly, though, Hamann reports only about six weeks later to Herder, that his "counter-critique to Garve's Cicero" (own translation) would have transformed into a "prodrom of morals" [Prodromum der Moral] (German original letter), i.e. into a groundwork.
Regarding a comprehensible story and more sources
I think the most readable summary of how things were and the philosophical drive produced by the reviews is in Eckart Förster's The Twenty-Five Years of Philosophy (2012), pp.48-53.
It is a good compromise between length, readability and academic quality.
There, it is made clear that Feder essentially mutilated Garve's review and added a comparison to Berkeley, which Kant so bitterly adressed in the Prolegomena. He received Garve's original review by mail (on Aug 21st 1783, see below), and found valid criticism of his arguments about freedom and morality, as Förster argues:
In the dialectic Kant had, on the one hand, shown that we cannot know anything about God and that theoretical cognition of supersensible objects must be ruled out as impossible in principle. On the other hand, he argues that certain propositions of practical reason cannot be true, or rather, cannot motivate action unless we can assume the existence of God and a future life. It is thus the validity and obligatory force of the moral law itself which reintroduces God into theoretical cognition, while at the same time it is the idea of God which serves to explain the bindingness and validity of the law . For “reason finds itself constrained to assume” the existence of God, Kant writes in the Critique, since “otherwise it would have to regard the moral laws as empty figments of the brain” (A811).
Kant is thus guilty of a petitio principii which only becomes clear to him through Garve’s objection (for the published version [i.e. Göttinger Anzeigen] of the review had passed over this point as incomprehensible). His explanation presupposes the very thing it is supposed to explain. (The Twenty-Five Years of Philosophy, p.52)
Förster is one of the main proponents of reading the Groundwork as essentially an 'Anti-Garve', even down to its structure, btw.
Also, together with Timmermann's commentary, a lot of sources can be found for further reading on the discussion.
For a different take with further sources, see Allison's commentary (Oxford UP, 2011), pp. 5-10 and 52ff. He takes a third stance, saying it is the challenge Garve's Cicero puts on "the very idea of a metaphysics of morals" (p. 6).
Good sources not mentioned in either of the commentaries, nor Förster (but only in German, just like the mentioned, excellent Schönecker, Beister, Reich, etc.): Immanuel Kant in Rede und Gespräch by Rudolf Malter (ed.) and Kant und das Problem des metaphysischen Idealismus by Dietmar Hermann Heidemann
Regarding Kant's own perception of the difference between the two versions of the review
Kant did indeed, in a letter to a third person, clearly express the severe differences between the reviews as published in the Göttinger Anzeigen and the Allgemeine deutsche Bibliothek respectively (to Johann Schultz, Aug 22nd, 1783)
I have the honor, dear sir, of transmitting for your evaluation the Garve review forwarded to me yesterday by Herr Oberconsistorialrat Spalding. I have only been able to skim it quickly, there being various other distracting tasks lying in the way; however, dispite his frequently mistaking my meaning, which is hardly avoidable, I found the review quite different and far more thought through than what is contained in the Göttinger Anzeige (which was supposed to be by him). (Ak. 10:349-50, translation taken from the Cambridge Edition Correspondence, p. 206)
Keep in mind that he writes this before he even had the time for an in-depth analysis of the arguments! Together with Förster's reasoning, it makes Timmermann's remark that "Kant still had little reason to be impressed" by the full review (xxvii) rather improbable.
As a counter-argument to that another letter from Hamann to Herder should be considered that deals with the Garve-review as well (from Dec 9th, 1783):
Kant is not satisfied with it and complains of being treated like an imbecile. He won't answer it; but he will answer the Göttingen reviewer, if the latter dares to review the Prolegomena as well. (translation from Corr., p.201, fn.1)
This translation is not entirely correct, though. Hamann originally writes "Er soll nicht damit zufrieden seyn", i.e. "Kant is said not to be satisified". Again - heresay. He points out the sentence before that he did not have the heart to ask Kant personally when visiting him, although this was the original purpose of his visit. One might argue that the contents of his reports are generally quite accurate, though.
Regarding a translation of both reviews
In case it helps for research: a translation of both reviews, as referenced in the Correspondence translation by Arnulf Zweig (pp.206-7, fn.1),
may be found in the appendices to James C. Morrison's translation of Schultz's Exposition of Kant's Critique of Pure Reason (University of Ottawa Press, 1995).