In case anyone is interested in running their fates with a LiPo battery, I have something that appears to be working after a bunch of trial and error.
I bought a MP2636 boost/charger board made by DFRobot for $10 CAD and 5000mAh LiPo battery (with build in protection circuitry) for $19 CAD.
Out of the box the MP2636 has problems when plugging the USB power into the board if the rpi is under moderate load — the charging circuit seems to steal too much current away from the rpi causing it to crash.
I modified the ISET resistor (pin 13) from 47.5K to 224k to limit the charging current to around 500mA, and, although that helped a bit, it wasn’t totally sufficient.
I then added a 470uF1000uF capacitor across the output rails and that seems to solve it. Now I can operate this like an UPS, switching between battery power and charging all while the rpi continues to run (even with most of the lights on my Arc lit up).
I think this battery will fit in the case, although I might have to make it ever so slightly deeper to get everything to fit.