This is a very common question here.
Current is pulled depending on the load, not pushed by the charger.
If the load needs or pulls 1A, then the power supply must be able to supply 1A. If it is capable of supplying more, then that's fine; the load will not pull more just because it's connected to a supply with greater capacity.
Think of it this way: A mains power outlet can operate a small space heater which may require 1500 watts. But it also can charge your phone. Your phone does not suddenly burn up or consume more power, even though it is plugged into a supply that is obviously capable of much more current.
Your computer's USB ports are only rated for 500mA (or 0.5A). If your battery pack requires 1A, then the USB port will be unable to supply the current. At best your computer will shut off the USB hub because of an over-current condition. At worst you will permanently damage your USB port(s).
I was treating this as a power supply and device rather than specifically as a battery bank and charger. In the latter case, the charger may necessarily limit current to a value that is acceptable for the battery bank composition. In such case, it is not recommended to use a different charger or power supply than what the manufacturer specifies.
Without knowing how your device operates, it is not possible to tell if current is limited at some other point, such as the battery bank itself, which would allow the use of a different power supply.
That being the case, I must recommend you adhere to the 5V 1A rating recommended by the manufacturer.