LiPo charger with LDO regulator
A typical LiPo battery can be charged with 1C however, will your USB supply this? If the device is only registered as a USB low power device, you should draw no more than 150mA from the USB or there is the danger that the port either shuts down or (best case) limits the current to 150mA.

Furthermore you are worried about that the battery is charged while your controller is running. If that's the reason for needing 1C charge current I would add some capacity to decouple the uC current draw.

It is correct that the linear voltage regulator will subtract the drop-out voltage from the input voltage. This voltage is depending from the current. You can find the applicable dropout voltage if you check the datasheet. Normally there is a graph showing dropout-voltage over current.