I agree that [name_u]Phoenix[/name_u] [name_m]Fitzgerald[/name_m] is a bit tongue-twistery, but it could work. I like the suggestions of [name_f]Phoebe[/name_f] and [name_f]Phaedra[/name_f].
More ideas:
[name_f]Nyx[/name_f] (“night”)
[name_f]Lux[/name_f] (“light”)
[name_u]Pax[/name_u] (“peace”)
[name_f]Beatrix[/name_f]
Viatrix (“traveler”)
[name_u]Alix[/name_u] (“noble”)
Phaenna
Photine (“light”)
[name_f]Fiora[/name_f] (“flower”)
[name_f]Fiamma[/name_f] (“flame”)
Another vote in favor of [name_u]Phoenix[/name_u] here. I know a little girl named [name_u]Phoenix[/name_u] and it’s pretty adorable on her. Also, I have a friend who grew up in [name_u]Phoenix[/name_u], AZ, and she speaks of it very fondly (and would very much like to move back). And I love the meaning. I also don’t think it’s too much with [name_m]Fitzgerald[/name_m], but that’s my personal preference, and I don’t mind a little matchiness between first name and last name.
I don’t see a problem with using [name_u]Phoenix[/name_u] in [name_f]England[/name_f], but it really is awkward to pronounce [name_u]Phoenix[/name_u] [name_m]Fitzgerald[/name_m]. For me it would be a deal breaker. I keep trying to say it, and I mess it up every time 
I’m sorry
[name_m]How[/name_m] would you pronounce [name_f]Xanthe[/name_f]? xx
I know I think this is the deal breaker for me also, OH don’t see the problem but that’s men for you haha - I love it so much but totally get what you mean when I say it in my head it sounds fine, when I say it out loud I get tongue twisted 
xx