Well the first thing I'll mention on the subject of potential imperfections of the KJV translation of the Song of Solomon is that I think "undefiled" is a bad translation in 5:2 and 6:9, those are the only verses that Hebrew word is translated that way, it's usually translated "perfect".
But what I mainly want to discus here is what I heard about recently of an alternate reading of the Song where Solomon and The Beloved are not the same person. And with that usually making Solomon a villain, maybe even a type of the Antichrist (1 Kings 10:14 does link him to the number 666).
One discussion of that possibly is called Solomon as the Antichrist. Another is one called The Shulamite Woman. The former has Solomon as only a small presence in the narrative, the latter has Solomon not The Beloved as the male voice talking throughout it, and thus manages to to use that to demonize much of the Sexuality in the Song. The latter view is totally illogical, clearly the male voice of the song is the Beloved. The former is an interesting theory.
Another site mentioning it is this one.
So I went over the Song with this in mind, trying to pay attention to things that might often be missed.
First I concluded that absolutely The Beloved isn't Solomon. In chapter 3 our Heroine is looking for her Beloved in the city, if the Beloved is the King the idea that he'd be so difficult to find is odd. Also in chapter 7 verse 5 the Beloved is talking and refers to the King as a clearly separate person. Likewise chapter 1 verse 12 has the woman talking about her Beloved and refers to the King it seems as separate.
What I'm not so convinced of however is that Solomon is a villain in the story, or in any way an alternate love interest. Chapter 1 verse 4 is what gives the impression that she was added to his harem, but it could mean something else, like a Princess becoming part of the court now that's she's of marriageable age. And the books of Kings and Chronicles seem to say Solomon only married foreign wives and concubines, while this heroine is clearly an Israelitess.
Overall I feel the references to the King/Solomon just tell us the woman is living in Solomon's house, and leaves at the end, seemingly receiving a dowry, the vineyards of Baalhamon.
It is often assumed the heroine of the Song has a humble background, often specifically a Shepherdess. There are Shepherding references, but none that prove that's the protagonist's background. It is equally likely the Beloved is the Shepherd.
In chapter 7 verse 1 the Beloved calls her a Prince's Daughter.
Chapter 6 verse 13 twice calls the heroine in the KJV "Shulamite". This is a bad translation, the actual -ites in the Hehrew text have no T, this ends with a Hebrew letter for "th", the one that like Heh usually makes a word or name feminine when used at the end. Don't be confused that "the" is used in the verse, in Hebrew and Greek the definite article is often used before personal names, it's just in English that it's considered grammatically incorrect to do that. This translation issue has caused people to imagine a location named Shulam that appears no where else in Scripture.
Her name is Shulamith. It's a feminine form of Solomon/Shlomo, which is why when assuming Solomon is the husband the name is sometimes taken as just poetically reflecting her as his wife.
Normally the feminine form of Solomon becomes in the KJV Shelomith (some males did have the name it seems, but it's first appearance in Leviticus 24 is clearly a Woman). In the Hebrew the only difference between how Shulamith and Shelomith are spelled is Shulamith has a Vav between the shin and lamed.
Vav is a letter that in time came to often be used like a vowel, and many scholars think after the captivity when the scribes became more concerned with representing vowel sounds that Vav and Yot started being used a lot more then they originally were. So I feel confident in concluding that Shulamith and Shelomith were the same name.
In 2 Chronicles 11:20 a Shelomith is the daughter of Rehoboam son of Solomon and Maachah daughter of Absalom. Rehoboam was 41 when Solomon's 40 year reign ended, and Solomon probably had Rehoboam at as young as 14 or 15. So Rehoboam's daughter could easily not only have been born but reached adulthood while Solomon was still alive.
Maachah, a wife of Rehoboam, is also called Michaiah daughter of Uriel of Gilead. The Hebrew word for daughter can mean granddaughter, in all likelihood she was Absalom's granddaughter, since Absalom died before Reboboam was born. And 2 Samuel 14:17 tells us Absalom had one daughter named Tamar (probably after his sister). This Tamar may have been the wife of Uriel of Gilead, which seems to be what Josephus implies in Antiquities 8:10:1. But we're also told Absalom had 3 sons but the sons aren't named.
Abijah also called Abijam, the full brother of Shelomith, died (seemingly of natural causes) after reigning only 3 years as King, and Rehoboam reigned only 17 years. We're not told how old he was, but he must have been more then 20, I think it's safe to say he could have been 20-25 when Solomon died.
Shulamith speaks of having brothers in the Song, Shelomith likewise had three brothers by the same mother, and numerous others. She would seem to be her mother's only daughter though, but not her father's, Rehoboam we're told had 60 daughters.
So I think it's possible that the protagonist of the Song of Solomon was Solomon's granddaughter, the daughter of prince Rehoboam, Shelomith.
Update November 2016: I've now done a follow up post on Typology and Symbolism.