5V Powerbank not turning on automatically when > 100mA are drawn
The power banks I've got don't deliver power until you press the button.
Press the button, and as long as the current stays above the minimum the power bank stays on.
If the drawn current drops below the minimum then the power bank shuts off and you have to press the button again.
In other words, measure the voltage at the power bank output. I expect you'll find it to be zero when the Pi fails to power up.  If the power bank isn't putting out any voltage, then you won't be able to draw any current from it.

This is why one project I am working on bypasses the 5V output and taps the cell in the power bank directly. To turn the power bank on again, I'd have to open the case and add something to &quot;push the button&quot; to get power on again.  That means an extra battery (like you have,) modifications to the power bank, and wires coming out of the power bank.
I found it simpler to just go ahead and bypass the 5 volt output and use the lithium cell directly with my own external regulator.
Make sure you connect to the correct side of the low voltage protection circuit for the cell (in the power bank) if you go this route.