Thanks Peabody:
Attached is that portion of my schematic. The MOSFET's are the wrong type, but that's the only part that I believe is incorrect. If I'm not mistaken the power will be routed through the power regulator (AMS1117), bypassing the battery portion entirely, due to the unlabeled MOSFET at the top.
The CP2102N has one mode where it reports there is only 100 mA available for charging - for low power devices. I haven't done the power budget yet, but I know my circuit will be drawing more than that, so I need to figure out if I want to allow it to run on battery, or just power down the components.
Thanks for mentioning the MCP73871. I'll give that a look.
Edit: I should mention that I'm currently investigating pairing a small MCU (PIC maybe) with a digital rheostat. I won't, of course, if some product like the MCP73871 will work. But, that's where I was heading...