Boost Converter shutsdown power bank
When you first run the boost converter, it may draw a huge current (Can go up to 10 times the nominal input current) for a short time (say, a few micro- or even milli-seconds) due to the "empty" input capacitor(s) (That is because an empty capacitor acts as short circuit. As it charges up, its resistance increases). That high current which flows until the input capacitor charges up is called "inrush current".

So, perhaps, the built-in (if any) overcurrent protection shuts down the power bank. Slowing down the start-up action of the boost converter may solve the issue:







simulate this circuit &ndash; Schematic created using CircuitLab

NTC has a high resistance at room temperatures. When you apply the supply voltage, powerbank's output sees high resistance so a small current flows. As the current flows, NTC's resistance decreases (because its temperature rises) so the current increases. That current keeps increasing until the NTC reaches its minimum value. This characteristic of NTC helps reducing the inrush current.

So, how to select the NTC?

Since your equipment draws 1A @ 9VDC input at max volume, your 5V-to-9V boost converter will draw nearly 2 Amps. So you should select an NTC with a current rating of min. 3A.