Of course the ц in мышца wasn't fitting the picture. Had it been a 16-17-century borrowing, why would a -к- in мышка be transformed into a ц? It was too late for the second palatalization to do this job.
Remembering this today, I decided to check in the Bible (I remember seeing a мышца there) — and confirmed that the word was present in the Church Slavonic version. Odd, I thought, but the Church Slavonic text is the Elizabeth Bible, published in the 18th century; maybe in an earlier Bible... Yet, sure enough, there it was in the Gennady Bible of 1499:
Мышца is listed in various dictionaries of Old Russian, some of them (Sreznevsky) giving it a more narrow definition of the muscles of the upper arm, brachium. At least brachium is the equivalent for the Church Slavonic мышца in both the Latin Vulgate and the Greek Septuagint. And it certainly makes sense regarding the подмышка.
Not a Latin borrowing then; but a much more interesting story of a word that co-existed both in Latin and in Common Slavic. Since it's rather unlikely that this unintuitive metaphor originated in different languages independently, it's probably suggesting a Proto-Indo-European origin of the word.