I also cringed when I saw the name.
Firstly, I dislike the tradition of honouring the royal family side of the family and not the mother's side - this goes for both Catherine and Meghan. I guess Catherine can say that Elizabeth was a nod to both the Queen and her mother (Carole's middle name is Elizabeth, as is Catherine's), but there's nothing in either of her son's names as a nod to her side of the family.
Secondly, it's interesting how times have changed in two years. When Archie was born his name was clearly not linked in any way to the royal family. Now they have shoehorned the name in, so that it honours the Queen and so they get to use their actual preferred name (Lily/Lili).
From memory, I don't think that Harry had to ask the Queen's permission to name his children, he was too far down the line of succession when Archie was born.
I just feel they have no sense or nuance or subtlety.
When they are sticking the knife in, they don't use a kitchen knife, they use a massive samurai sword.
When they are trying to ingratiate themselves with the royal family, it's love bombing central.
I am going to give them the benefit of the doubt and assume they spoke to the Queen about it before hand - because it feels so personal to the Queen, especially just after Philip died, that I'd find it really inappropriate if they hadn't checked first.
Hopefully this is a bit of an indication that they think they may have over stepped the mark slightly with the OW interview and the apple documentary, and are trying to pull it back a little. Again unsubtle, but a step in the right direction is always a good sign.