Syntactic analysis:
Main sentence: American herman melville chose to write his own tragedy.
Adverbial of Time: Two and a half centuries after Shakespeare wrote Hamlet,
Rational adverbial: Inspired by the great figures and poetic power of English drama, this is a past participle phrase, not an independent structure. Its logical subject is the subject of the main sentence: American herman melville.
But as a novel with a whaling ship as the background as a supplementary explanation, it can be said to be ellipsis = but written as a novel with a whaling ship as the background.
Two and a half centuries after Shakespeare wrote Hamlet, inspired by the great character and poetic power of the British dramatist, herman melville of the United States decided to write his own tragedy, but this is a novel with a whaling ship as the background.
By the way, chosen is a past participle. Is it wrong? We should choose it and use the past tense as the predicate verb. Otherwise, there is no predicate verb in this sentence.