They claimed they were not given adequate time to respond to developments in the trial, which they said involved a number of legal errors.
They further submitted that expert evidence of two veterinarians called by the Crown should have been excluded because one was biased against them and the other was in breach of disclosure obligations.
The couple, who had applied to be discharged without conviction, also said their sentence was excessive.
The Court of Appeal, which released its decision today, upheld the earlier District Court decision and dismissed the appeal, with the exception of the order to pay court costs.
It found that the order was not just and reasonable, for reasons including the Balfours being unable to pay it.
Half of the animals were put down as a result of the ill-treatment and about 50 cats had to be treated by a veterinary surgeon. Some of the cats took up to three years to recover fully from the abuse.
During the couple's sentencing in Palmerston North District Court in June last year, Judge Grant Fraser said he accepted that the pair were of good character and that their offending "came about as a consequence of circumstances".
Nevertheless, it had a major impact on the animals, he said.
"There can be absolutely no doubt that this is serious offending of its type.
"Your behaviour was not wilful but fell well short of the standards expected, and animals suffered. It is necessary for the court to make you accountable for the harm you did the animals."