Lawyers acting for David Gilroy, 50, told the Court of Criminal Appeal in Edinburgh that he had been the victim of a miscarriage of justice.
Gilroy, formerly of Silverknowes, Edinburgh, was jailed for life at the city's high court in April for murdering Suzanne in May 2010. The office worker – whose body has never been recovered – went missing following the May Day holiday that year.
Prosecutors believe Gilroy dumped Suzanne's body close to the Rest and Be Thankful road near Arrochar in Argyll.
Yesterday, the lawyer acting for Gilroy, John Scott QC, told the appeal court that police acted illegally when they interviewed his client in the days following Suzanne's disappearance.
Mr Scott also argued that Lord Bracadale, the judge who presided over Gilroy's trial earlier this year, acted illegally when he did not stop proceedings after some jurors saw a medical report which speculated about the circumstances surrounding the book-keeper's disappearance. However, judges Lord Carloway, Lord Brodie and Lord Wheatley rejected Gilroy's appeal saying that the police acted legally when dealing with him and that Lord Bracadale acted correctly during his trial.
Lord Carloway said: "This court is of the opinion that this appeal cannot succeed."
Now Gilroy's legal team plan to appeal to the UK Supreme Court in London.
Gilroy's wife Andrea sat in the public benches just yards away with her husband's father.
Mr Scott argued that when police interviewed his client in the days following Suzanne's disappearance, they contravened his right to have a fair trial.
The QC said police interviewed Gilroy as a witness to a potential crime, but Mr Scott said evidence showed they suspected him of committing a crime.
The lawyer said detectives did this to deprive him of the right to consult a solicitor before speaking to them.
Mr Scott said this resulted in Gilroy providing officers with evidence that was used against him at his trial. He said this evidence should not have been heard.
Mr Scott also said that officers forensically examined Gilroy's body at the police station.
Mr Scott added: "The full forensic examination given to him suggests that he was something more than a witness."
But the appeal court judges told Gilroy that they had to reject the appeal.
Mr Scott asked for leave to appeal to the UK Supreme Court. Lord Carloway told Mr Scott to hold off asking for leave until he and his fellow judges wrote a report detailing their reasons for rejecting the appeal.