Loudly is Complement or adverb/adjunct in the sentence The cat / scratched / loudly
Your answer is correct.
Complements can be either subject complements (in which case they follow a stative verb) or object complements (in which case they follow the direct object). A subject complement is required to complete the verb. In the example they give, it's a transitive verb and nothing is required to complete it. "The cat scratched" would be a complete sentence, so "loudly" is an adverb in "The cat scratched loudly".