How to keep powersupply/step-up module citcuitry "awake"

https://www.richtek.com/assets/product_file/RT2808A/DS2808A-00.pdf

That does not look at all like a powerbank controller. It's a buck converter so it can only produce voltages below the battery voltage (2.95V to 4.2V). The recommended input range is 5.5V to 36V. The HOTCHIP HT4928S appears to be a MUCH closer match.