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A big factor in the optimum focal length of Fresnel lens would be the beam angle of the flash you are using. A flash that has adjustable beam width, and is set at its longest setting, may indeed be a good match for the standard Better Beamer lens. I suggest you buy a replacement Better Beamer lens for $10 so you have it as reference, and can compare it to the other lenses you play with. The link at B&H is: http://www.bhphotovideo.com/c/product/683113-REG/Visual_Echoes_FX_FL_Fresnel_Lens.html
A reflective housing to try to capture side light and redirect it in to the Fresnel is not going to help much if any. For light to contribute to the beam, it needs to originate from near the focal point of the lens where the main flash tube and built in flash Fresnel are located. The side light rays will reflect back in toward the Fresnel, but will be coming from too much of an angle, and will just end up focused down range off of the main beam.
The over all length of the unit really does not matter much, so your search for a shorter focal length Fresnel may not have much benefit. The Better Beamer comes apart quickly for transport, so the longer length is not really an issue. A larger sized Fresnel might capture more of the beam angle of the flash, so that might help out with more power down range, but the specific flash and Fresnel size and Fresnel focal length will all enter in to finding an optimum.
The Better Beamer works really well as is, so improvement may not have a big payback. Another factor to consider is that if you put too much light on the bird, you may end up harming it temporally, which is not good. Some photographers will not even consider using beamed flash or any flash on birds because of possible harm. My thought is to keep on using beamed flash occasionally, but to cut the power down some depending on the range to the bird. I also try to stay up with factual information about possible harm, but tend to ignore rumors that are not backed up with true science.