If the price is right, of course it would.
They'd have to be stupid to pass on the opportunity to pick up all that backlist and active contracts. Life-plus-seventy copyrights are valuable.
(As for regulatory approval, since the "Big 5" aren't all that big in terms of market share and APub is only a midsize publisher they can easily secure regulatory approval. Note that APub books aren't intentionally --or literally--Amazon exclusive. They are available to all vendors via Ingram, including ebooks. So neither post-merger size nor verticality is likely to be much of an issue.)
Which BPH would they buy?
Well, since most are foreign-owned, the only one likely to ever be available is S&S, which is owned by CBS and controlled by the Redstone family, which is having a bit of top level turmoil.
http://money.cnn.com/2018/02/01/medi...lks/index.html
The way the BPH pecking order breaks down, PRH controls almost half their market share, Murdock's HarperCollins, a quarter, S&S a third, Hachette, a sixth, and MacMillan the rest. Profitwise, though, MacMillan is reasonably profitable (for a publisher) while S&S is marginal and Hachette is breakeven at best.
So again, the best candidate for a sale is S&S. Which makes up about ten percent of the US market. (Hachette is in dire need of better management but it'll be a cold day in the nine circles when the French allow Amazon to buy even a small disposable piece of Lagardere.)
Again, no likely objection to the hypothetical sale.
Now, what would they do with the BPH?
While it would surely be tempting for Amazon to fire all the S&S staff and move the work to Grand Rapids, Michigan with the bulk of APub, Brilliance, etc. (Or maybe to HQ2) the big value of S&S to Amazon would be their political influence-buying operation and their distribution channels. After all, B&M bookstores can afford to boycott APub books (3-4% of the pbook market) but they could hardly afford to boycott S&S plus Apub. (~15%)
So their most likely changes would be:
1- improve their backend tech and online marketing (like Bezos has done for the Washington Post). They would finally have a good website.
2- increase their operating budget (like Bezos has done for the WP), especially for buying politician memoirs
3- switch their baseline contract terms, for new deals, to Apub terms (reportedly a 50-50 split of net on ebooks).
4- no change of management structure initially. Amazon tends to give plenty of autonomy to businesses they acquire as long as their performance meets expectations. So it should be business as usual for a while. Of course one would expect the rest of NYC publishing to ostracize S&S staff and freeze them out of the "normal" day to day collusion lunches and gatherings so there would be a fair amount of attrition among the executive suite, improving financials notably as the golden parachute gang "self deports".
5- Agency has been very, very good to Amazon so I doubt they would abolish it openly. But since the publisher sets the price and S&S would have access to Amazon big data, they could set the prices to the most profitable price points and use Agency as a tool to gain S&S market share. The others would either have to follow suit or abolish Agency altogether. Which might drive Apple out of the ebook business.
So yes, I can see Amazon buying S&S from the Redstones. It wouldn't impact tbeir bottom line much (buying RING is a better use of their cash.) but it would annoy the heck out of the other BPHs. A worthy goal all by itself.