I am not Steve (obviously) but the way I handle that scenario depends on the bird. If it’s twitchy and restless, I do the stills at high shutter speed and take the iso penalty.
If it’s a more patient bird, I go to a lower speed, take a couple stills, go back to higher shutter speed and wait. And i keep on hoping to the birding gods during the whole process that the bird is actually patient
Thats why I can’t wait to try the Sony A1. You can set one back button focus for BIF and the other for Stills. That includes all the parameters (shutter speed, aperture, high iso limit, AF mode, frame rate... whatever you want to change). It’s like having 2 (or more) cameras set completely differently at the press of one single button. At least on paper that seems the perfect solution to the problem you describe and we have all run into.
Apparently the camera also remembers what bird you had focused on for stills (likely with a very small focus area for accuracy on the eye) and jumps back to that same bird when you change back button and resume with your BIF settings and a wider AF area... now that sounds too good to be true but I can’t wait to try.