The Hebrew phrase in the Song of Songs is poetically short and concise: “Ani l’Dodi v’Dodi Li”
אני לדודי ודודי לי
(It’s also specifically talking about a male “beloved”, but the English does not carry that particular idea. Nor does the Klingon I am about to suggest.)
You can adjust the literal meaning just a teeny bit and get a similarly poetic and short phrase in Klingon:
bangwI’ ghaH ‘ej bangDaj jIH
He/she is my beloved, and I am his/her beloved.
It says the same thing from a slightly different angle. The possession is elevated from an elided genitive to a complete sentence, and the love-labeling is folded into a possessive suffix rather than being a copula.
Klingon does not really handle genitives the same way that English does; you can’t say “I’m yours”, but rather you have to specify “I am your (something)”.
So, for example:
bangwI’ bangna’ jIH ‘ej bangna’wI’ ghaH bangwI”e’.
“I am my beloved’s definite beloved and my beloved is my definite beloved.”
Of course, that’s a bit tautological (at least the second part).
This begs the question: What do you mean by saying that your lover is “mine”? You probably don’t mean “my property”, nor “my acquaintance”.
Perhaps you mean that your partner is completely loyal to you, and you to them?
bangwI’vaD jImatlhchu’ ‘ej jIHvaD matlhchu’ bangwI’.
“I am completely loyal to my beloved and my beloved is completely loyal to me.”