Hi Andrew,
There’s a good to fair chance you need more voltage, not more current. It’s likely that little plugpack is not capable of the same standards of line and load regulation as the official Pi supply is because they’re designed for different purposes.
If you want/need to use a power source other than the official one, your safest bet is to assume the official supply specifications are the same as the device requirements (5.1V ± 5% ie. 4.85V - 5.36V for a Pi).
If your supply is dropping below 4.85v (which is actually within the USB specification) you’re likely pushing the limits of what the Pi can operate on.
There’s a good discussion of this issue here:
Sometimes if I need a wider gap on my power board, I find a double adapter like this is quite handy:
