Just a concept, sort of a block diagram. I use a very similar circuit but with alkaline cells and no charging circuit; it's worked well for me. For input I use a 5V regulated wall wart. The diodes are 1N5818 or similar. For the regulator I use an MCP1640, which is limited to about 300mA output. For the circuit to work, two conditions must be met: (1) The battery voltage must be less than than the wall wart voltage, and (B) The input voltage to the boost regulator must be less than the output voltage. A LiPo battery with an appropriate charging circuit should work as well.