5V UPS design IC/ battery-backup circuit selection
So to cope with the voltage drop I would recommend adding a boost converter behind the MOS-FET. It will be fine if the 5V of the usb input go through it and will also improve the stability of the voltage it outputs.

For the MOS-FET you should try to get one that has pretty much the same characteristics as the one in the schematic. Most important values are the gate-source voltage Vgs and the on-resistance vs. Vgs, because if these are different the MOS-FET might not fully turn on/off.

Since I've actually taken the above mentioned circuit and improved it a little bit. You can check out the whole circuit schematic and a video about it over at instructables. Maybe you can reuse my circuit. Heres the schematic: