LDOs that handle more current than the MCP1700 tend to also have higher dropout voltage. But you might consider using two MCP1700s in parallel to provide the additional current but still keep the low dropout. I've seen this done in several commercial circuits, but I don't know if it's an approved practice.
To have the charger properly terminate charging, you don't want there to be any load current flowing through the charger. The solution is the standard "load sharing" circuit, consisting of a P-channel mosfet, a schottky diode, and a resistor. It has the external charging source powering the load directly whenever it's plugged in, while charging takes place independently, if needed.
