I am trying a new technique I picked up on this forum. When shooting BIF I set the camera to manual and set my shutter speed and F stopwhere I want them. I then set ISO on auto. Also use a mono pod with a ball head. Just started doing this last week last week. Seems to have increased my keeper rate but I must admit there is more PP involved. I cannot access exposure compensation with this set up on my Rebel T5. Not to worry though Have a used 7D coming this week and from what I've read you can use EC with this set up. BIF are addicting, something I never shot in the film days. Shoot and have FUN!

*With the right post-processing software and settings* the white balance of your camera DOES transfer to the initial view of an image when you open it for post-processing.

In other words, If I open a raw image in my camera manufacturer's software, or if I set the Camera Calibration in Lightroom to the same Picture Style I set at the camera, the software will correctly interpret the EXIF data from the JPEG preview stuffed inside the raw file wrapper. It will display the image using very close to the same parameters as the camera used (or would have used) for a JPEG saved there. That INCLUDES application of the white balance set in the camera.

THAT'S why I always recommend folks get the WB as close to right as possible at the camera. If you perform a Custom White Balance at the camera, you can make minor tweaks in post-production, because the initial view of the image will be close to perfect (ASSUMING you have a properly calibrated and profiled monitor). This is most helpful when you are in an environment where the light is unchanging. It is also most helpful to those who plan to record JPEGs and do NO post-processing.

If you're not using a ColorChecker Passport and its subsequent workflow, including a proper white balance target in a test exposure of each new scene, at the camera, can be used to achieve "click white balance" in post-production. Record raw images, and the click balance tool will get you very accurate white balance, using this method.

Whenever I've been confronted with a situation such as you describe, I'll meter something stationary in the same light as the bird would be. My hand works well held over my head and metered. That would simulate the shaded side of the bird in flight. As long as the lighting is similar to that on your hand, there will be no difference in the exposure between your hand and the bird.

If the lighting is changing, broken clouds blowing by, take two meter readings. One when your hand is in shade, the other when in sun. Remember those two settings. Then just switch between the two depending on the lighting, as there is little time to do much else.
--Bob

Raw + JPEG is a sound strategy.

If there is a particular important or valuable image that would benefit from retouching and or enhancement, then the raw file would be superior than working with a jpeg. While a JPEG can be improved, working with a raw file is essentially like re-exposing the image. Because a raw file is a digital negative with much more image information by a factor in it, pixels that appear the same color, aren't, but the difference may be very subtle… this subtle difference can be accessed by soon who knows what they are doing, to improve an image with detail that a jpeg simply no longer contains. This isn't opinion, this is indisputable mathematical fact. Of course it depends on if it was properly exposed to begin with… a poor image is a poor image, but a poor RAW file is way more fixable that the JPEG version.

As ALL cameras take raw files, then apply automatic, irreversible adjustments and then compress the image using the pixel averaging JPEG algorithm. So include it, stating they are duplicates of the JPEG he can see, and that if any warrant retouching… the digital negatives are available for a professional to work with.

This leaves the option open… perhaps never to be used… but having JPEG & RAW files represent a value added exchange at no extra cost for you, if the need should ever arise.