Can a "who" act as both a pronoun and a conjunction at the same time?
Solution 1:
Who identifies the person you will sue as the subject of the verb in the relative clause, murdered. A conjunction cannot do that. A pronoun can. A conjunction simply connects two separate clauses.
A relative pronoun is a pronoun used to mark a relative clause, and having the same referent as the element of the main clause (usually a noun or noun phrase) which the relative clause modifies.
You will notice that dictionaries don't have an entry on who as a conjunction.
Solution 2:
It’s a relative pronoun, and nothing else.
Solution 3:
According to SIL, "a conjunction syntactically links words or larger constituents, and expresses a semantic relationship between them", and a pronoun "is a pro-form which functions like a noun and substitutes for a noun or noun phrase." It seems that both definitions are satisfied by the use of who in the OP sentence. Interestingly, SIL does not define the term relative pronoun in its glossary of linguistic terms.
Who has both pronomial and conjunctive properties that can function simultaneously in sentences like I will sue the person who murdered my neighbour, and many others. In fact, it is probably used this way more often than not.
Who are you?
Who stole my beans?
I am who.
Sarah is who.
It's pretty common for who to only exhibit its pronomial function as in these examples, but I think it is more common that it exhibits both properties.
I want to know who stole my beans.
It is Sarah who stole your beans.
In both of these sentences and many more like them, who is referencing a known or unknown entity, joining two clauses in a specific relation, and/or adding information to a noun with relative clause. These relations are both pronomial and conjunctive at the same time.
In terms of how you 'treat' the word in a sentence like this, I think you have to acknowledge that it is doing both operations, but in terms of how to correctly parse it, that is another question that people will no doubt enjoy arguing about. From what little I know about various parsing systems, they don't really allow for multiple functions like this, so who knows? All the relative pronouns seem to do this, and when I explain this to my ESL students, I just point out to them that these words are doing both of these things, and they really don't have any difficulty with that.
https://glossary.sil.org/term/conjunction
https://glossary.sil.org/term/pronoun
https://www.collinsdictionary.com/dictionary/english/who