Woosh
it's not true that most batteries on sale with kits don't use the most energy dense cells. Her kit can be bought with 17.5AH HL battery which costs £100 more than the same with 13AH battery.
Oh I'm afraid that's not what I found when I was in the market. I was looking for a 48v battery in the smallest case available. There were numerous batteries available with lower densities, larger cases and less than optimal sizes. There were few sellers, for small case batteries around 14ah, they usually didnt have stock worst of all really difficult to get detail information. There was a bit more availability for 36v but still not great.
The 17.5AH HL battery uses the Samsung INR18650-35E cells
As said 36v had better availability but again most suppliers seem to focus on lower density cells and thats where the availability was. It might have improved a bit by now though,
After a while I gave up on down tube cases, mainly because of lack of information from suppliers more than the availability of better cells. Also even in the smallest cases the suspicion was that size of the packs overall were still larger than the equivalent Bosch (even using similar or sometimes better cells) but that info even if true was of no real practical use other than to say it was possible.
Woosh
The fundamental problem with fitting a smallish battery like the small frog (223mm x 120mm x 134mm) is of fitting.
The size did seem adequate for the use proposed. However it looked less than Ideal in terms of the configuration of the bracket. I did think if you could sort out a bracket the general shape might lend itself to being a handle bar mounted battery.
Woosh
You need at least 120mm clearance between the saddle and the rear mudguard to fit the battery to the seat post below the saddle.
Riley's saddle has to be set so low that she could not even have that space on her bike
I presume you mean below the saddle to the top of the mudguard wheel. The simple answer is dont try and put the battery under the saddle do something like this mount to the rear of the saddle.
This particular rack probably wouldnt work as the geometry is less than Ideal for a small frame and you have to imagine the bag as a battery but it illustrates what I mean. I would prefer a conventional rack. You might need geometry similar to the Topeak small frame beam rack to work as well.