I don't claim to know who said what to who and when, but when I first heard of the choice of name I cringed.
It was a toddler's nickname, and to anyone with any sensitivity, is clearly an intimate family matter - not the sort of thing which ought to become public property. My children had many nicknames (too twee to mention here!) and it would be embarrassing and intrusive if someone used them and linked them to their original owners.
I can't remember when Lillibet Sussex was born (sorry, cba to Google) but if it was after the departure from the UK and particularly if it was after the Oprah interview, then I'm absolutely stunned at the audacity of their thinking that name would be OK with QUII. What on earth were they thinking? If they wanted to honour the Queen why not just call the little girl Elizabeth, which is not so deeply personal?
And yes - as another pp said - why not Doria?