I think this depends on accent.
In Australia, I’ve only ever heard CLAR-uh (first syllable rhyming with [name_f]STAR[/name_f])
[name_u]CLAIR[/name_u]-uh sounds awful to me. I wouldn’t use the name if I thought it was going to get that pn. 
Australian. CLAH-rah.
I think you’re right - Americans say [name_f]CLAIRE[/name_f]-ah and Brits/Aussies say CLAH-rah.
In English (USA), I’d pronounce it as [name_f]Claire[/name_f]-uh
In Spanish (Mexico) I’d pronounce it Clar-ah
I’m from [name_f]Scotland[/name_f] and I say CLAR-uh, I’ve only ever heard it that way!
I automatically pronounce it Clah-ruh because when i was younger i had a friend who said it that way but [name_f]Clara[/name_f] is actually a very common name in my family tree and I know that at least my great-aunt pronounced it [name_f]Claire[/name_f]-uh (not sure about all the others) and we’re all English so the [name_f]Claire[/name_f]-uh pronounciation is definitely around in [name_f]England[/name_f] 
Clah-ra. Had it been spelled Clarah, I’d prn it like [name_f]Sarah[/name_f]. I’m not a native English speaker, but I thought the ‘h’ changed the pronunciation (like in [name_f]Mariah[/name_f]).
I’m from Australia and I pronounce it Clar-uh. I’ve never heard anyone say [name_f]Claire[/name_f]-uh: makes it an entirely name.
[name_m]Just[/name_m] like the name [name_f]Tara[/name_f], which I discovered when visiting US Is generally pronounced Ta-[name_m]Ra[/name_m], but in Australia, [name_m]New[/name_m] Zealand & [name_m]Britain[/name_m] it’s [name_m]Tah[/name_m].-rah. It’s my sister’s name and she said while we were in US “I have always loved my name, but in [name_u]America[/name_u] I HATE it!”. Amazing the difference an accent can make!
I’m from Australia, and i say it CLAR-uh.