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Hey there, and welcome to Balmung! 
I honestly don't know the specifics on how present timezone wise some of the other FC's are, but I know Azeyma's Oath has an EU presence as well as the Mischief's Coven. Both of these FC's are actually next-door-neighbors too, so there's some bleedover between them for RP, but they're both radially different concepts. The Oath is a treasure hunting and relic recovery FC, whereas the Coven is a front for organized crime (aka "the mob"). There's also the Guiding Light, which I believe also has an EU presence. Theirs is more the pursuit of knowledge across Eorzea, and is a relatively new startup (so you have some opportunities to help something grow).
I might be a little biased for the Coven (*cough*), but you can't go wrong with the Oath or the Light!
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I honestly don't know the specifics on how present timezone wise some of the other FC's are, but I know Azeyma's Oath has an EU presence as well as the Mischief's Coven. Both of these FC's are actually next-door-neighbors too, so there's some bleedover between them for RP, but they're both radially different concepts. The Oath is a treasure hunting and relic recovery FC, whereas the Coven is a front for organized crime (aka "the mob"). There's also the Guiding Light, which I believe also has an EU presence. Theirs is more the pursuit of knowledge across Eorzea, and is a relatively new startup (so you have some opportunities to help something grow).
I might be a little biased for the Coven (*cough*), but you can't go wrong with the Oath or the Light!