Dreading another trip to the cat clinic, her veterinarian

'her' has no clear referent. Same with 'she'. A,B and E are out.
Between C and D, C is better.
The noun 'Jessica' clearly states who 'her' is referring to. The phrase broken out by commas 'a rather combative two-year-old black male' modifies the cat.

C.

Dreading another trip to the cat clinic, her veterinarian was persuaded to treat her cat at home, a rather combative two-year-old black male.
A)...
###Incorrect modifier. "Dreading <blah>" should modify "Jessica/her", not "her veternarian"

B) She persuaded her veterinarian to treat her cat at her home, the cat being a rather combative two-year-old black male.
###Missing antecedant - Subject of "she" is missing.

C) Jessica persuaded her veterinarian to treat her cat, a rather combative two-year-old black male, at her home.
###Correct

D) Jessica persuaded her veterinarian that her rather combative two-year-old black male cat would best be treated at her home.
###Passive voice "cat would best be treated"

E) She persuaded her veterinarian that her home should be the place that her rather combative two-year-old black male cat would was treated at .
###Totally messed up with excess verbiage.

(C)

Dreading another trip to the cat clinic, ... Who was dreading another trip to the cat clinic? The answer to this question would give us the first word of the remaining part of the sentence.

Examining the options, only (C) and (D) remain. (D) can straightaway be eliminated (too wordy and ambiguous usage of "her"). That leaves us with (C)

I also pick C
Seems it's a general agreement on C

I simply love the way how E is said. He told him that he should treat his brother as if he is his own brother. Oh boy. :-D

But yes I believe C is correct. And Kapslock, I believe that ETS does prefer an active voice to a passive one.