Single cell with a boost converter and a micropower microcontroller to give you the ability to have an adjustable setpoint will work. Just use a common boost converter to 5V and then use the micro in sleep mode with a slow clock to sample input cell voltage every so often so that as the cell reaches say 25% of capacity it drops the output to low battery level, then at 10% it turns off the output to simulate flat without damaging the lithium cell. The cell protection circuit will stop discharge at a point well after that, but the boost converter will still allow unregulated cell voltage through, less the Schottky diode drop in the converter, till that point, so will keep any RAM alive in the device till the protection drops out the output. Makes charging easier as well, you can use a slightly modified charge circuit and a common 1 cell charger, available off the shelf to add to the cell.
Otherwise just go buy a USB power bank and use the board off it, it has every part aside from the voltage variation inside a single small package, and will work. the boards are also available cheap on fleabay, with either USB ports input and output, or with only the board and no connectors for integration into your own equipment, just add a protected cell to it or a non protected cell and a ultra cheap cell protection board, which is also available in 10 packs.