To deal with the semantics issue, I assume you're actually referring to the Low Voltage Warning; I'm not aware of a Low Power Warning - though in this case, they are effectively the same thing.
The Low Voltage Warning is generated by some version of the 7704 chip on your RPi 3B (e.g. MxL7704 on the RPi4B). I can't say what version is used on the 3B because the RPi Organization won't tell us. So we will stick to some generic information for the answer here.
The 7704 monitors the 5V input, and when it falls below 4.63 volts, the Low Voltage Warning is triggered. I've done enough testing on this to feel confident in saying that, **"You are actually experiencing a Low Voltage Condition at the "
5VSYS" pin of your 7704. I say this in spite of the vendor specs you have quoted, your battery condition, etc, etc.
I'll assume you want to know why you are experiencing this Low Voltage Warning. The "usual suspects" are well-known:
When you switch your RPi to "ON" (connect power), there is a very brief, but measurable, time when it is drawing above specified current... think of it as an inrush condition when all the filter caps on the 5V Bus must be charged.
Your battery pack also employs a voltage regulator to get from battery voltage (~3.7V) to +5V. We have no specs on the regulator, but we may rest assured it follows the laws of physics, and cannot respond instantaneously to a change in the current demanded by your RPi. IOW, there may be a time during start-up when your regulator is out of regulation, and this is enough time to trigger the Warning.
Batteries have internal impedance that limits the flow of current to the regulator. This will act to further limit the regulator's ability to respond to the added current demand during start up, or anytime the load changes abruptly.
The wire leads from the regulator/battery pack to the USB micro connector on your RPi have resistance and inductance proportional to the wire diameter and length. Using the largest possible wire gage, and the shortest length will help - but not necessarily eliminate - this contributing cause to dropping below 4.63V.
You're using a charger instead of a power supply. A charger has a different regulator than a power supply... it's designed for charging - not for accommodating rapid load changes.
Hope that helps your understanding.