The logic of the Academy was centered around Aristotle, and that of the Stoics inspired by Zeno. Given the close association of Stoicism with the elite of the Roman Empire, and the eventual adoption of Christianity as the state religion by Rome. Why then wasn't Stoic Logic adopted by the church - is this because the church adopted the logic of the Academy to put clear water between themselves and the Pagans, ie the Stoics?
I'm not sure the church ever adopted any "official logic". Thomas Aquinas was an Aristotelian of the so-called Thomistic variety and since the Catholic Church adopted much of his philosophy in its doctrine it might be called "Aristotelian". But I'm not sure that they ever adopted an official formal logic of any sort.
The logic of most of the medieval logicians was also a good bit more sophisticated than Aristotle's and quite similar to the Stoic's modal logic. For instance, Ockham was the first to give a complete modal syllogistic logic (the Stoics had a flawed one and Aristotle gestured at one).
Check out The Development of Logic if you're interested in the history of logic. It's a bit dated since it was written in 1962 but it's still one of the best surveys of the history of logic.
Regarding the prevalence of Aristotelian thinking, I think this is largely historical accident. Kneale reports (as you note in your question) that Stoic and Epicurean logic were dominant around the time of the Academy, with Aristotle being relatively ignored. The source of Aristotle's medieval popularity might be Boethius. Boethius translated a bit of Aristotle's more logical works, he is generally the source of the Medieval's knowledge of Aristotle's logic. Prior to Boethius, however, the sense I get is that Aristotelian Syllogisms were not nearly as popular as the logic of the Stoics. Another factor is that Boethius's presentation of Aristotle made salient the "problem of universals" which became a major topic of debate in the middle ages and prompted further study into Aristotle.
But even the work of Boethius (who was active during the late 5th century/early 6th century) went beyond simple Aristotelian syllogistic logic. He had his own propositional logic, one presentation of which is a commentary on Cicero's treatment of Stoic logic. There is an old book in the series Studies in Logic and the Foundations of Mathematics (amazingly edited by L.E.J. Brouwer, E.W. Beth, and A. Heyting) called "The Propositional Logic of Boethius" by Karl Durr that contains a pretty good presentation of Boethius stoic-style propositional logic (including his theory of modality). The book is really a beast of historical and curious notation, though, which makes it a bit hard to read if you aren't already acquainted with some historical literature (in particular it utilizes both Polish notation and Russell's notation from Principia Mathematica).