Hi Jack.
Light is your friend as you can stop down then, f8 ish seems to work well. A lot of variables come into it though as mentioned above and it will always be a compromise, to some degree.
I'm a bit OCD about it, I like the whole bird sharp but even a goldfinch, can be sharp on the head and eye (the important bit) but the tail can be OOF.
A trick I've learned is to get the bird perched side on and parallel to the sensor. The problem with that is you lose some dimension or depth but it can work.
One thing I would say, focus on the eye, that has to be sharp, everytime.