Looking for Power Bank with "Always-On-Feature"
Lithium cell based powerbanks use a boost converter to boost the voltage from the Cell to the 5V for the USB. They turn off because keeping the voltage boosted when not used wastes quite a bit of power.

If all your devices can run straight on 3.3V you can use a Lithium cell directly with a voltage regulator (and with the normal protection circuitry).

Otherwise you can simulate a button press on the on button with a gpio pin to activate the boost circuitry when you are ready to take your measurements; let he voltage stabilize; use transistors to connect the components to power; take your measurements; disconnect the components and then let the timeout expire after which the voltage will drop again.