Apologies for the awkward phrasing; I'm not a linguist so I'm not really sure how to phrase it. Also apologies if this would be more appropriate for r/linguistics, but I'm more interested in the historical context of the development.
To clarify: there's a trend in English where you can, for instance, replace the word "you" with "your ass," such as with sentences like "That guy got his ass beat," or "Your dumb ass is gonna lose a few teeth." The synedoche (listen I don't get a lot of chances to use that word and dammit I'm gonna ride this train as far as it takes me) feels very intuitive to use, but out of context it feels odd. If somebody's hindquarters have teeth, I don't want to know about it. On top of that, it feels somewhat strange as a whole in that it practically feels like a pronoun unto itself. "Your ass," as a replacement for "you" feels like it doesn't place that much emphasis on the buttocks as symbolic of that person. "The White House" as a replacement for "the president," by contrast, feels like it blends in the building, the office, and the person to a much greater extent.
I think it's generally more of a grammatical trend in AAVE, but I've also seen it used in the context of media set in the American South and/or the military (in a drill-sergeant, "get your sorry ass over here, maggot" kind of way), and the linguistic lines between those three settings feel a bit ambiguous at times, so it's difficult to pin down. Where and when do we start seeing this trend take off and get more widespread?
It'd be more appropriate for r/linguistics, yes. Historical linguistics (language change and etymology) is a part of linguistics.
Technically it's not synecdoche (where a part is used to describe the whole), it's a functional pronoun, and techncially it's not just a pronoun, it's an intensifier. It forms part of pronominal phrases (your ass, his ass) or part of APs (crazy-ass, dumbass), and can be used reflexively or non-reflexively. It does a few other syntactically-interesting things. For an examination of its use and productivity, there's a squib (a brief paper) here, if you have access (either there or from scribd). I can't find it freely available, but emailing the authors will usually get you a fair-use copy. Linguists despise putting a price on this sort of thing.
As for its history, that's harder to answer. To my knowledge nobody has done a point-by-point examination of the changes required to bring about each of the current usages of ass. What we do have are dates of first appearance in publication, which I stress are not necessarily indicative of when a usage began, just when it became acceptable enough to finagle its way into print. Most of them are US military sources, but that's not enough to claim it originates within the military - much more likely it existed regionally for a longer period of time, became popular military slang, and became multiregional only after discharge. Certain of its more colorful combinations may have been military institutions, however. The transition of ass from a donkey to one's backside is similarly attributed to 19th century nautical slang.
As an intensifier those dates and instances tend to center around the period following the Korean War, though some isolated instances occur earlier. raggedy-ass (1930); bigass (1945); badass (1950s); dumbass (1959); smartass (1961); hardass (1961); candyass (1961). These words appear as modifiers first and only several years later as nouns. There is a limited German analogue arsch- as an adjectival prefix.
Pronominal use that's fully productive appears in print only by the late 1990s (the first squib I mentioned uses examples from comments about the OJ Simpson trial) (although I found a Fawlty Towers usage of it dating to the 1970s). It may have existed before then, but I don't know anyone's put the sources together. The fact is the study of constructions like this doesn't usually extend to its historical origins, even where that information is theoretically available, because most linguists are much more interested in studying its relationships with theory, which require live subjects and current usage to test.
Although at least one paper identifies it as an AAE (AAVE) construction, this use certainly also exists in multiple regional dialects of American English.
One interesting area of inquiry in the ass camouflage construction is universal third-person agreement, even with first or second person referents: Your^1 ass must be kidding {your/it}self^1, where the referent is clearly second person, in verb forms demonstrating agreement: Your ass {is/are} dreaming. This suggests ass has a unique structure from pronouns like self, either with the possessor originating outside the DP or as evidence of a parasitic head. That's kind of cool.
We know it has other limitations, including prosodic limitations. It's been compared to the -fucking- infix, which requires prosodic feet on either side (that's why Minne-fucking-sota is good, but Min-fucking-nesota is bad). Ass doesn't have quite this same restriction, but it does worsen as an attributive when no other morphemes follow it.
A realistic order of operations for the evolution of ass would be:
Beavers, John, and Andrew Koontz-Garboden. 2006. A universal pronoun in English. Linguistic Inquiry MIT Press.
Collins, Christ, Simanique Moody, and Paul M. Postal. 2008. An AAE camouflage construction. Language, 84.29–68.
Elgersma, Diana. 1998. Serious-ass morphology: the anal emphatic in English.
Levine, Robert D. The ass camouflage construction: masks as parasitic heads. Language, Volume 86, Number 2, June 2010, pp. 265-301.
Sidiqqi, Daniel. 2011. The English intensifier ass. Snippets, Issue 23, May 2011.