He's been ordered to have the dog destroyed because it's dangerous but he refuses to comply. What is the role of the "to have the" in the sentence and how it is separated from he's been ordered to destroy the dog?
Here, have is a complex transitive verb that has a direct object ( the dog) and an object complement (destroyed).
In English grammar, a complex transitive is a verb that requires both a direct object and another object or an object complement. (ThoughtCo).
An object complement is a noun, a pronoun, or an adjective which follows a direct object to rename it or state what it has become. (Grammar-Monster)
Look at this example sentence from the second reference, showing the object complement "spoiled". You'll see that spoiled is not a verb; it is an adjective.
We consider fish spoiled...
In your example sentence, "destroyed" is an adjective and an object complement. (The direct object is "the dog".
That leaves the answer to you question, which is that "have" is a complex transitive verb.