"Carrier Falcon Princess". Not a BL but FL also become a bird. It's really similar.
Not bird, but there are BLs of various other pets the MC isekais/transforms into lol:
Horse - Shouwaru Boukun Kishu to Nagasare Senba
Cat - I Have to Work Hard as a Cat Today, Blackcat and Magician, My One and Only Cat, Ninkyou Neko Choko
Dog - Inu mo Arukeba Koi ga Hajimaru, Wolf in the House, ch 3 of I'm in trouble because I'm loved so much in this different world anthology
Snake - The Snake Demon Wants to Flee
Unicorn - Cherry and Unicorn
The manhwa of this is only promotional but if you are alright reading novels, The Northern Grand Duke's Hamster
this is similar but fl turns into a crow: https://www.mangago.zone/read-manga/i_ve_become_a_crow/ (she's really pretty!!)
Any1 have a similar story to this? (づ￣ ³￣)づ