Your suggested setup would work, but you Arduino would only have about 3.2 to 4.2V to work on - there is no boost circuit to make your 5V supply.
You can get cheaper power banks that are not "intelligent" and thus have no lower limit of current that you need to draw. Look for a power bank with no button to turn it off, or one where the button specifically is to show that remaining charge and nothing else.
// Per.