Hi, will this power the ThruNite catapult mini safely and at full power output for the torch?

Yes it will. The Speras PB11 is very similar to the battery supplied with the Catapult Mini, quite possibly the same celll inside.

Torch companies do not make the battery cells. They add protection circuits and branding in most cases.

The cells are usually made in Japan or Korea by companies like Sanyo, Panasonic or LG.