How cute and what a lucky girl!
If you are comfortable with her going bareback there are many pads that have a very pronounced hand grip which might aid her in staying on.
Also, my personal preference for children... wintec has really cheap lead line kids saddles that has an adjustable fit (velcro panels) so it should work on your tubby appy or any future ponies? Lol :) Also, the hand grip is more of a bar which I think kids have an easier time holding onto vs. a saddle horn.
Have fun with your daughter!