do you have a storyboard and stills of the scene and especially the hat available?
with composites the elements needed will depend a lot of which element is in front of /behind which element and how practical or unpractical it is to key or rotoscope and how much the perspective changes.
Without seeing photos I am highly sceptical that you would have any benefit of shooting the hat separately, the actress will probably move so much with it that you can't imitate its movement in the clean plate. And if the bird flies from behind the hat (the hat is first in front of the cgi element and then the cgi element goes in front of the hat plate) you will need either bluesceen the actress WITH the hat which would probably be very unpractical OR rotoscope the corners of the hat which are in front of the bird at the start of the shot (I would recommend this approach) .
Are you on fixed camera or can arrange a camera setup which has no parallax error when panning/tilting at the start of the shot (if the camera is not dollying in any direction) ? if so, you can shoot a clean plate of the background WITHOUT the actress or any extras/actors on the path of the flying bird. then the actress with the bird hat IF the corresponding hat corners can be rotoscoped easily if the bird is flying behind them. if the decorative elements can also be rotoscoped easily it may be easiest to just shoot those two elements and rotoscope the bird behind the decorations when it lands.
but that depends on a lot of variables and you may need to 3d track the hat's movement when the bird is on it to get the bird's perspective change to appear right.
I recommend shooting on high quality camera+format and deteriorating it after the vfx is done to get the cellphone camera look.
What your vfx supervisor said about the scene? as said it depend on how the bird flies and how the hat moves and especially the hat elements colors compared to the background colors and each other (to determine if rotoscoping is practical or not or if blue/greenscreen is needed) .
(for example if you have feathers etc. very difficult-to-rotoscope elements on the hat and almost similar color background, you may need to use chroma/greenscreen and rethink the vfx)