Flaw In Chamber Of Secrets ??  

Interesting point Fawkes is a phoenix though and is therefore immortal, and even if harmed would just regenerate

As I remember, Fawkes didn't look in to the eyes of Basilisk when she fights it, because she attacked its eyes and blinded it in the first moment she fights it.
It sounds like Fawkes know how dangerous Basilisk is and destroyed its most powerful weapon as soon as possible

yeah.. it's like fawkes flew on top of the basilisk and butt facing it clawed the eyes off.. remember fawkes claws are more near his/her butt that face.