What a great question! I can tell you think deeply about English grammar. The answer is, you are both right! The important thing to understand is two terms: 'form' and 'function'. For example: "I like you." In this sentence "I" is a pronoun but it's functioning as the subject of the sentence. "You" is also a pronoun but it's grammatical function is object. So in your sentence, "whose" is definitely a pronoun. But in that particular sentence it also has the function of introducing a relative clause (also known as a adjective clause, one type of subordinate clause.) Because it has this function, it can also be referred to as a "subordinator." It's form is 'pronoun'; it's function in this particular sentence is 'subordinator'.

It's an adjective clause not a relative pronoun. If you want to use a pronoun in the sentence (it will be THAT not WHOSE )