I love the sound of Tigris, too, and I like the imagery/feel of bird names. For me,personally, Tigris and Egret would be strictly middle name territory, though I wouldn't classify Tigris "unusable" as a first name. I would not scoff or joke if I met an Egret; I would just really hope she likes her name and hasn't been given too bad a time over it. Aside from Egret being an unusual name choice, it's also similar to the word, "regret", which may or may not bother you, or her. I'm thinking either name would work best paired with something "classic" or "traditional" or "historical" in feel...
Tigris
AntoniaTigris
DorotheaTigris
HelenaTigris
HonoraTigris
JosephineTigris
LaraTigris
LenoreTigris
MaryamTigris
NaomiTigris
PaulinaTigris
TheodoraTigris
Victoria
Didn't do too well for you, but maybe these will spark some ideas. :)