Cheap and easy way to power MCU from Li-Ion, and keep it powered during charging
One part that immediately comes to my mind is a MAX1555, which is a charge management chip for a single LiPo (or Li-Ion ) cell. It too is designed to operate directly from a 5V source, and even has a separate 5V USB input with a lower charge rate. The reason I thought of it is that it has an output designed to drive a LED when it is powered and the cell is found to be in need of charge. So when the cell is fully charged it will turn off the LED. You might be able to use that LED output to disable the regulator. I'd also like to point out that I've actually used that chip very recently in an application similar to yours. In my case I do have a user operated power switch, but I've found that the MAX1555 still charges the cell perfectly even with the switch on, because the chip works by sensing the cell voltage, and because the charge current it supplies is significantly more than the MCU. That may or may not be the case for you but it might open another option. In my case the only effect was to make my charge cycle, normally about an hour, take just a few minutes more.