A few thoughts.
The Mail's argument that the letter was intended for publication is clearly irrelevant to the copyright claim. If I write a novel and the Mail gets hold of a copy of the manuscript before it is published, does that allow them to print large sections of it without my permission and without compensation? Of course not.
Meghan's failure to inform the High Court that her former comms secretary provided some information for an unauthorised biography is, in my view, irrelevant to the case. It is likely that this information would have been part of disclosure had the case gone to trial, but the judge stopped the case before it got that far on the grounds that publishing her private letter was clearly excessive and unlawful. It seems highly unlikely that any co-operation she or others around her gave to an unauthorised biography would have altered that judgement.
She will not be prosecuted for perjury (and nor would anyone else in this situation) as, in order to get a guilty verdict, the prosecution would have to show that she knowingly gave false information as opposed to giving information that she genuinely, but mistakenly, believed was true, and that this information was significant to the case. In my view the prosecution would fail on both fronts.
Why are Associated Newspapers trying to overturn the verdict? Is it because they think they will win at trial? I doubt it. I really don't rate their chances even if the Court of Appeal does send it to trial. The Mail are doing this because they want the circus of a trial so they can use it to try and damage Meghan's reputation. They hope that some of the mud they throw at her will stick so that they can sell more newspapers. After all, that has worked well for them up to now.
I am sure their lawyers would have told them prior to publication that they could face a claim for invasion of privacy and breach of copyright, and that they would probably lose. They went ahead and published anyway because they thought the opportunity to destroy her reputation would be worth it in sensation and increased profits. They want to destroy Meghan so that they can profit from it.