There is a difference between being grammatically correct and being in common usage. If you have to argue with someone about your language choices then the real problem is that you are not communicating effectively.
The problem with very is that it is primarily used as an adjective modifying other adjectives to indicate scale and scope. Most English language learners try to use it as a standalone indication of large or important. As a result, many will use phrases such as, "she is the very girl" instead of she is a very feminine (or beautiful) girl".
For American usage then:
The "very girl" is specifying that she meets what you are searching/hoping for. It fits but is not going to be understood by many "average" people as these sort of references are not used in common speech.
The other two examples will be considered wrong by those you speak with due to failure to establish context.