The SQ11 can be disassembled by taking of the multicolour strip, then removing the tiny Philips screw.
I de-soldered the Li-po battery, soldered positive and negative leads to a (totally overkill) 18650 protection circuit.
I then verified that this quick hack still recorded video and charged the battery (very slowly, as the SQ11 motherboard is tailored for the 200mAh battery). As the 18650 is fully removable, I can fast-charge it in another contraption if I want.
Finally I got through 15 iterations for the 3D printed case, discovering the hull command in OpenSCAD and taking full advantage of it.
Unfortunately I discovered that there are multiple version of the SQ11 with different camera mount and heat spreader, so if you buy one, I can't guarantee that my case will fit unmodified.