The load is not that high I think.
How is the battery connected to the pro mini?
A phone battery charger will expect to see a demand for 0.5A from the phone - substantiallyless than this and it will assume the battery is fully charge and switch itself off (to save itselffor the next phone I presume)The moral is use a phone charger battery pack only for charging phones.What you need is a simple battery pack of a suitable voltage.
Time to break out your maker skills. You could crack open the case, get to the power leads from the battery, and wire them to your own 5V regulator.