Just doing some small math, I
think it might be possible to fit
two of these
Lithium Ion Polymer Battery - 3.7v 2500mAh batteries in case, along with a pi 3, all side-by-side-by-side, giving 5000mAh at the same 3.7v.
I thought about maybe stacking them on top of each other, but I took some measurements where if hypothetically you took out the battery holder, that would give you 0.488" of height to work with. The batteries are 0.300" thick, and this is not including the thickness of kite's controller board (and any electrical tape insulation that goes with it

) so that won't work.
However, I measured from the right post that supports the D-pad, all the way across to the post that supports the Y-button, and that gave me about 6.5" with a little room to spare. The batteries are 2" in width, so two batteries would take up 4" in width, and the Pi 3 is 2.3" in width, so all together that leaves you with 0.2" of wiggle room to distribute across the batteries and the Pi 3, not bad.