5. Fearing another visit to the cat clinic, her veterinarian was persuaded to treat her cat at her home, a rather combative two-year-old black male.
(A) her veterinarian was persuaded to treat her cat at her home, a rather combative two-year-old black male
Incorrect. It is not "her veterinarian" who is fearing another visit.(B) she persuaded her veterinarian to treat her cat at her home, the cat being a rather combative two-year-old black male
Incorrect. "being a rather..." should be close to "her cat". (C) Jessica persuaded her veterinarian to treat her cat, a rather combative two-year-old black male, at her home
Correct, nice appositive.(D) Jessica persuaded her veterinarian that her rather combative two-year-old black male cat would best be treated at her home
Incorrect. Persuadate to or into, not that. (E) she persuaded her veterinarian that her home should be the place that her rather combative two-year-old black male cat was treated at
Incorrect. "was treated at" doesn't make any sense. Persuadate to or into, not that.