I wouldn't be surprised if they just gave her a heads-up that they were naming her Lilibeth, I very much doubt they asked if they could, that's not really their style.
They were so resentful and aggrieved about not getting the half in/half out, I really doubt there was much affection behind their use of the name, I think they were protecting their kids future connections to the RF, the RF being their golden goose (said golden goose now appears to be dead as a Dodo).
They have proven themselves to be a snarky couple when they get push back.
"While there is not any jurisdiction by The Monarchy or Cabinet Office over the use of the word "Royal" overseas"
""Following conversations with The Duke, The Queen has written confirming that in stepping away from the work of The Royal Family it is not possible to continue with the responsibilities and duties that come with a life of public service," the palace statement said."
In response, Prince Harry and Meghan pushed back with a statement that pointedly referenced the Queen's: "We can all live a life of service. Service is universal."