... the power bank... ... expects larger load, ...

Externet, This power bank is meant to be used to charge cell phones and has circuitry to shut off if it is not outputting (x) amount of mA.

That explains it. You are trying something the power bank is not meant for. What is the '(x)' value ?

Load can be increased by repeating the circuit in post #12 as many times as needed to overcome 'x' .
So far, seems you want a rechargeable LED lamp. Your approach is very convoluted and expensive.
----> https://www.aliexpress.com/item/200..._1&btsid=7ba0a549-5942-4bff-8360-1a1d18a01bfb

----> https://www.aliexpress.com/wholesal...3958&SearchText=solar+rechargeable+flashlight

 

In other words, two 200 ohm 1/8 watt resistors in parallel would act like one 100 ohm 1/4 watt resistor, if I'm correct.

Correct.