A few objections against Solomon being the bridegroom are that Solomon was a king, not a shepherd (1:7), that he had hundreds of wives (versus 6:9, remembering that Solomon's first wife was an Egyptian princess and cannot have been the Shulamite), and that he was not a good role model for marriage (whether or not we take the book allegorically). Are these or any other objections strong enough to say that Solomon is not the bridegroom in the Song of Solomon?
The Song of Solomon is attributed to Solomon as the author according to verse 1. It is clear to scholars if Solomon is actually speaking from experience. It is more likely that he wrote about the idea romance and the people are supposed to be archetypes. The name of the bride is "Shulamit" (7:1). It has the same hebrew root as Salomon and therefore it is suggested that she it can be translated with "the one of Salomon". It is the female Version of the name Salomon. It can be seen as the bride belonging to Salomo and that he is the actual bridegroom. Or that they are both archetypes like similar to the use of man ('ish) and woman ('isha) in Genesis 2:28 ("[...] she shall be called ‘woman,’ for she was taken out of man.”)
However the rabbis say that the book symbolizes the relationship of GOD with Israel and the church alternatively said that it symbolizes the relationship of GOD with the church.