I found a battery/UPS module for the Pi made by Sunfounder that was relatively cheap.  The more DIY options were going to boost the price, and I'd rather have something off the shelf if my son is going to be messing with it.  Overall, it seems to eliminate my low power issues which I suspect are due to trying to power the screen over the Pi's USB port.  Wiring only involves some USB cables for the most part.  

I can attach the power switch to the EN and Ground terminals to kill power to the machine when it's shutdown and have some options for adding LEDs for power status and charging status at some point.  I'll probably need to reprint something and add some holes to make that work, but not until a future revision.