This is an interesting idea, but it seems to me that you already have a 3.3V regulator. So what you need is a circuit that connects the battery to the RAW input when USB is not connected, but which simply disconnects the battery from RAW when USB is connected. Then you would connect the V+ USB pin (UVCC on my schematic) on the Micro back to the V+ input of the charger.
So when USB is powered, it would supply both the charger and the Micro's UVCC. But since the battery is disconnected from the Micro, it would continue to charge independently to completion. And of course when USB is not connected, the battery would supply the Micro's regulator.
This is what's called a load sharing circuit, which permits using the device while properly charging the battery. It can usually be implemented with a P-channel mosfet, a shottky diode and a resistor. And I think the diode is already there on the Micro.
Take a look a Microchip's AN1149 application note. Different charger, but same idea.