I think you’re probably correct all around.
And if HM herself/the Palace is fed up, I do find that understandable.
Yes, I can see if either not being informed at all or being told that they were using Lilibet not Elizabeth as a fact was the straw that broke the camel's back. It's very personal.
Especially as it would have been Prince Philip's 100th birthday tomorrow and that's sure to be a very poignant day at best.
The fact that no sources are correcting anything suggests that the Queen wanted this out there and that however she found out, she wasn't supportive.
All this drama over a baby's name at what should be a happy time. I really wonder what the end game is in threatening to sue the BBC and any other media that reported on it. Control? Forcing a statement from HM? Because any publicity is good publicity?
In months to come most of this would be forgotten. Yes there will always be those that point out how ..ironic it is to name the baby Lilibet after trashing everything royal in the last few months or talking about privacy and using a private nickname but well that's what comes of using this name. But most would move on and just print whatever cute stories about Archie and Lili you fed to your sources.