Well. It depends whether you see the institution of the monarchy as having gravitas and a certain aura of propriety.
If one sees the Windsors as just a family, then I suppose that the pros and cons hinge on whether Her Majesty herself would be offended.
But she isn't just a private person, she embodies leadership and the stability of the nation state.
But frankly, isn't it like what a friend of mine once mistakenly did: she was a teacher, and stuck a piece of paper on the notice board in school, with a message about dates, only it was the one for staff and signed by the Head using his familiar name, and meant for the staff room, and not one for the pupils to see. This was a big faux pas, letting the kids see his personal name like that. He wanted to retain gravitas in order to execute superbly his job of service to that school community .
If you publicise the pet name of the monarch - and it will definitely be publicised by your giving the reason you've chosen that name for your new baby - then you do rather deflect from the dignity of that office of state. You make something public which is supposed to be private, and which is usually used by the named person to create intimacy and safeness for themselves as part of a little safe haven to return to when their risky and high-stake work is over for the day. There is a little bit of vulnerability about continuing to use a young child's take on your name. It's saying to your loved ones: "With you, I'm just that young innocent girl - myself: just me".
It's not the baby having the name that is do ambiguous in terms of constitutional etiquette, but the fact that the rest of us know a little secret - and not really anyone else but the monarch's private little humour to share.
Part of the authority of monartchy, that which is used for the public good, depends on monarchs conducting themselves in a reserved and proscribed manner, at all times. Let's not forget, that under the constitution, if parliament failed - for some reason to do with lots of ministers resigning, or something - then the monarchy would be asked to lead and make decisions for a time. Shining a light on their personal lives detracts a bit from the institution's formality.
You don't get barristers calling the Judge by his nick name in court. It's all formal to reflect the seriousness of the business they are conducting.
Even many everyday people only invite certain intimates to call them by a special name. Let's not forget, too, that a name you wish to be called by when a young adult, might not remain the private name you want lots of distant cousins to use as you grow older.
If the Queen can stand all this intrigue and drama, then she must be a very strong person. I hope she does have vast reserves of fortitude.
Yes, if William and Catherine wouldn't dream of uttering the Queen's pet name, then they won't want to say it in connection with the new baby. But ironically, re names, we are speaking of somebody who isn't called Kate, but let's herself be referred to that way out of modesty, good humour, and not wanting to make a fuss, when her role is to shine a spotlight on those volunteers who do good works.