I would say the non-hyphenated version flows better, unless you want to go by both [name_f]Emma[/name_f] and [name_f]Claire[/name_f], then I would say Emma-Claire.
Are [name_f]Emma[/name_f] and [name_f]Claire[/name_f] both first names (like a double barrel), [name_u]Taylor[/name_u] a middle name, and H. a stand-in for your last name? [name_u]Or[/name_u] is [name_f]Emma[/name_f] the first, and [name_f]Claire[/name_f] and [name_u]Taylor[/name_u] the middles? [name_u]Or[/name_u] is [name_u]Taylor[/name_u] a last name?
If you’re planning on going by [name_f]Emma[/name_f] and [name_f]Claire[/name_f], introducing yourself as Emma-Claire, then I’d go by the hyphen.
But if you just want to go by [name_f]Emma[/name_f] and use [name_f]Claire[/name_f] as one of your middle names, then [name_f]Emma[/name_f] (first name) [name_f]Claire[/name_f] [name_u]Taylor[/name_u] (middles) H.
Would be better
Good luck on your legal name change!
I know how hard it is to live with a name you don’t like, (I use a pseudonym on Nameberry) so it’s amazing that you’re doing this
I’d say, do you want to be called [name_f]Emma[/name_f] or Emma-Claire? If you just want to be [name_f]Emma[/name_f], I’d go with [name_f]Emma[/name_f] [name_f]Claire[/name_f]. If you want to be known by both names, I’d choose Emma-Claire