Bob,
The charging circuit is part of the 3.7v to 5v inverter.
I soldered (welded ;-) ) an old Motorola really thin cell battery to mine and it charges without issue.
Q: when you apply power via the USB, is there a blue flashing LED?
When there is no battery or the battery is low, it flashes.
Mine stopped flashing and stayed ON after my battery was charged.
Post a photo of your board (remove the rear cover) so you can compare to mine.
73 Larry