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I can see the appeal, and I like it for its Greekness and its birdness. However, I probably wouldn't use it because of its cuteness. I like Phee as a nn, Bee is cute too but I'm not really a fan of the B sound (especially when the alternative is a soft pretty sound like Ph).
Belphoebe on the other hand, has all of the coolness and none of the cuteness. It would definitely age well and its nicknames could be Bel, Bella, Belle, Bell, Phoebe, Phee, Bee, etc. A Belphoebe could even be just Phoebe most of the time. This is definitely less usable as a full name, however.