The 6D is well known to handle low light superbly, so as long as you're not cropping too much and not underexposing, I'd have no hesitation in going to ISO 3200 when necessary.
What focus mode are you using? For perched birds I suggest you go with a single focus point and ensure it lands bang on the eye. If you look in your first shot, compare the sharpness in the feathers between the eyes to those grey ones around the right shoulder (as we look - the birds left). If you look at the shots in DPP4 you can have the software show where the focus point landed. If the point is always landing on the eye but the focus is still off, then you'll need to consider micro adjustment of the lens, but don't alter anything until you're sure your technique is spot on.
I always recommend staying in AF servo (continuous) as opposed to single shot, as even a slight movement, either by you or the bird will result in the shot being soft otherwise.
You also need to up the shutter speed, 1/200th will give a very low rate of keepers with small twitchy birds. Up the ISO and deal with the noise in PP, at least you stand a chance of having a sharp file to start off with.
Mike