Geh?! Horsebird? How do they pronounce that? There are too may ways:
umatori (which would be horrible, but most likely)
bachyou (second most likely, prefix of horse with suffix of bird used in compound kanji names like that)
(unlikely, but possible candidates)
matori
taka
tato
tatotsu
banka
banto
bantotsu
meka
meto
metotsu
maka
mato
matotsu
In fact, they all sound pretty horrible. I'd almost rather go with chyouba (birdhorse) to be closer to the original chocobo.