Thanks cantsin - good to know.
Obviously it would be better if the batteries lasted longer. For example, shooting a piece in a jungle/desert/remote area etc. for a week would require many more batteries (even if the physical area/weight is smaller than the equivalent of powering a DSLR/FS100/whatever). As a ratio, like Andrew said, it seems like the pocket camera batteries perform well.
A Sony NP-F770 battery is about the same height as the pocket camera. It would only need to be deeper to accommodate the battery, making it look more like the letter 'L' from a bird's-eye view. This could also provide a better hand grip.
But at this point beggars can't be choosers. Still interesting to talk about.