My name is Claire. I've only gotten the "Hello Clarice" reference once in my life when the last Hannibal movie came out. Other than that, I always get people commenting on how pretty my name is. I know that is not 100% the same, but, I just thought I'd share. I don't think a lot of younger children will know about the "hello Clarice" thing, so, you wouldn't have to worry much about peers using that until at least middle school. And, that is if they even stumble across 'The Silence of the Lambs'.
I think Clarice is a pretty name. If you are not too fond of it, though, what about another form of the name such as Clarissa or Clara?