I have a lot of opinions on this idea of “needing” a formal name. I have a nickname name, and whilst I want to change it, I never felt like I was missing out by not having the full version. I don’t think having the full name would’ve made me any less likely to want to change it, because I don’t like the name at all!
Also, here in the UK, nicknames as full names is just the way we name. You’re more likely to meet an [name_f]Ellie[/name_f] that’s just [name_f]Ellie[/name_f] rather than [name_f]Elizabeth[/name_f] or [name_f]Eleanor[/name_f] “Ellie”. Names like [name_m]Charlie[/name_m], [name_m]Bertie[/name_m], [name_f]Tilly[/name_f], [name_m]Alfie[/name_m], [name_m]Archie[/name_m], [name_f]Millie[/name_f], [name_f]Katie[/name_f] etc are, I believe, mostly more popular than the full names. I have never been taken less seriously because of my name. I think it’s becoming a bit of an old-fashioned perspective to think that everyone needs a formal name. Plus, if she grew up and wanted to be a [name_f]Charlotte[/name_f] “Charley” there’s nothing stopping her changing it herself!
I think you should name your baby something you like - if you like the nickname, but not the full name (in this case your husband not liking Charlotte), then I think it makes more sense to use the name you like and will actually call the baby (i.e. [name_f]Charley[/name_f]) than to use [name_f]Charlotte[/name_f] just for the sake of having a full name. Also, if you named her [name_f]Charlotte[/name_f], she might be a [name_f]Lottie[/name_f] (or just [name_f]Charlotte[/name_f]) one day and not a [name_f]Charley[/name_f] - might be worth thinking about whether you’d be okay with that.. I would only say to use [name_f]Charlotte[/name_f] as the full name if you both loved it, but since that’s not the case, I think [name_f]Charley[/name_f] is perfect for you 
I like [name_f]Charley[/name_f], and this is my favourite spelling for a girl! I like [name_f]Lilah[/name_f] too, but I think from what you’ve written here it feels like [name_f]Charley[/name_f] is your winner