Hi Stef,
Just to say that I agree with John and Binnur, and think it would be a lot easier if you could put some more distance between yourself and your parrot. Perhaps setting up in a long hallway would be possible?
BIF are always challenging to photograph, and even more so when they are coming straight at you especially when in such close proximity (requiring a faster shutter speed) with a bird cage behind you. I've captured large birds coming straight at me in close proximity at f/8 and SS 1/1600 but the background is typically sky or water, much easier to do than what you're trying to do. If you can manage more distance between you and the bird you'll be able to use a larger aperture. Your focus looks great to me, nice and sharp on the birds head and eyes. Personally I like the motion blur seen when used with lower shutter speeds on fast flying birds.
I've never tried anything similar to what your trying but if I were to do so I would practice by trying to capture the BIF from a side position, first (easier), and I would set up my background prior to trying to nail the shot so grabbing and keeping focus on the bird was easier. (no competing objects) I've no experience with flash so I don't know how that effects your choice of settings.
Anyhow I imagine what you're trying to do is extraordinarily difficult and that more space, taken from a side angle (before the head on shot but then again you've managed the focus so perhaps not) with your unobstructed background set up, while you practice might be worth a try. Or if 100% safe for your bird take him/her outside?