"Pull up" resistor refers to a resistor connected from a signal or power line to a higher voltage positive supply. Without disassembling the unit you have access only to the power line and ground. But a "pull down" resistor from power line to ground (i.e., across the output) would consume extra current and if sufficient it would keep the power bank on. Of course, the power it consumes would be wasted but this wouldn't be a big deal if you could recharge the power bank regularly. I tested several power banks and the amount of current required to keep them on varied quite a bit. In general, higher capacity units required more current to stay on.
When looking for power banks that would work, I came across a number of DIY circuits that would load the output with a low duty cycle pulse. This wouldn't waste as much power but adds considerably more complexity.
The hearing aids charge in about 3 hours, so if you have a cell phone or other higher current device that takes longer, you could get an adapter that allows you to plug them both into the same power bank port and let the cell phone keep the power bank awake. Just remember to set the hearing aids so they don't automatically turn on when the power finally goes away.
I have a couple of other USB-charged devices that draw very little charging current, too -- an MP3 player and keychain flashlight -- although neither as little as the hearing aids.
Roy