Most Vets specialise in 2/3 animals most. They are usually reluctant in case there is a mishap in their treatment of the animal (if it's not their speciality) but all are capable of wound treatment in any animal. Fingers crossed x
Poor little chicky. I know you said your vet doesn't treat birds, but all Vets are trained in pretty much every animal ( my uncle owned his own practice) they will be able to sterilise and stitch (if neccesary) so if you are concerned maybe pop down anyway. fingers crossed for you and your chicken.