Inexpensive SMALL UPS circuit

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:

TL431 is working out of the specifications, the datasheet states a minimum 0.7mA to 1mA cathode current required for the reference to work properly, see the tables listed from page 5 to 13 "minimum cathode current for regulation" parameter.

At a first sight R1 is way to high even before the voltage will be cut by U3. Also the cathode voltage must be at least close to the reference voltage, see the comparator example in page 21 and the table in page 22 and also your common sense on how a reference should work.

Maybe lowering the value of R1 and feeding it from the highest voltage source through two diodes might do the job.

^{simulate this circuit – Schematic created using CircuitLab}

If your circuit works well on battery and you're worried about the higher current consumption then you could make a compromise and slightly modify the schematic to supply TL431 in parameters only when V1 is higher enough.

^{simulate this circuit}

Update

I couldn't make your circuit work as is or with slight changes.

The voltage drop detector doesn't work as intended since M1 transistor is always open when U1 gets in the working range with cathode voltage higher than 2V.