Beatrice, nine, sent a letter to the actor asking for piracy lessons to help lead a mutiny against the teachers.
What does the asking participle phrase act as? Why is participle phrase used instead of multiple clause?
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It acts as a subordinate clause further explaining the action of the sentence (in which Beatrice is the subject, sent is the verb, the letter is the direct object and the actor is the indirect object). I don't understand how you're posing "participle phrase" and "multiple clause" as opposing alternatives; the participle phrase is one of the multiple clauses in the sentence.
The entire sentence is quite grammatically correct.