I'm probably the only person who doesn't like the name Lili/ Lily. I do, on the other hand, like Lilibet very much. It's pretty, it's unique and more elegant than Lili in my opinion. That's my opinion on the name by itself, without the context...
However, for them to use this name after spending the past year, and recent months in particular, trashing the Queen's parenting, her family & life's work, is beyond hypocritical and distasteful. I cannot perceive them as genuinely trying to honour someone, to whom they have not shown much respect at all, and whom they have caused a tremendous amount of hurt during one of the most difficult times in her life.
If they had shown the Queen love and respect as Harry's grandmother and a dear family member, I would think it a sweet tribute. In this context, I cannot.
I also really, really doubt that they asked the Queen's permission, which they should have since it is an intimate nickname and certainly not a common one! And she is a Queen, so in my opinion they should have asked permission even if they had wanted to use Elizabeth as a first name (as opposed to a middle name as the other members of the family have done). It's just the respectful thing to do.
If one disregards the disrespectful behaviour and constant seeking of fame (and that's what this name is meant to do, in my opinion, provide more fame and money making opportunities) of the parents, I think it will still be a sweet connection for the Queen and Lilibet to share. The Queen can't be happy with what Harry & Meghan are doing, but she might still be happy to have her nickname live on in her great-granddaughter. And when Lilibet is old enough to understand the Queen's life and reign, I hope it will be very special to her, too