「あなただけの私には...今の私はなりたくない。」
means the same thing as:
「今の私は、あなただけの私にはなりたくない。」
「Xになりたい」 ("to want to become X") and 「Xにはなりたくない」 ("to not want to become X") are both very simple constructs and the 「に」 and 「には」 MUST be used respectively for the phrases to make sense. Since 「なる」 is an intransitive verb, you need to use a particle in introducing the indirect object "X".
Let us now replace the X by the actual words 「あなただけの私」("I that exists only for you" or "only yours" to borrow your TL).
The speaker is saying that, currently, she does not want to become the person described immediately above.