Battery ratings; am I reading this correctly?
Theoretically, and 8000mAh (8Ah) battery should be able to deliver 8 Amps for one hour, but the actual usable capacity will depend on how much current you attempt to draw, relative to the Ah rating.

Lead-acid batteries are normally rated assuming a 20 hour discharge.  If you attempt to discharge one in one hour, you may only get 50% of the rated capacity.

I assume that your battery is a Lithium type, since you mention a protection circuit - in that case, the maker has decided on a maximum safe discharge rate, and you attempted to exceed it.

You should study the datasheet for your battery to see how much current you are permitted to draw.