One other thing soldier ... assuming you can handle the truth ...
When you find it necessary to use a higher then desirable ISO setting (because it is the only way to increase your shutter speed) to almost ensure that your chosen bird is not suffering from motion blur then resist that temptation. Shooting at an auto setting, such as “sports” does not let you intervene. At the “Av” setting you are in control. If you decide that f6.3 (wide open) provides an adequate depth-of-field then you set that and the shutter speed will adjust itself and you can see that in your viewfinder. If that shutter speed seems too low, even with “OS” on-board, try to resist cranking the ISO to really high levels, shoot slower but do so in short bursts. Chances are that at least one shot of that burst will be sharp despite the slow shutter speed and that sharp image will have all the benefits of a lower ISO (less noise, more colour latitude, wider exposure latitude, more tonal detail and therefore less severe contrast.
As you were.