We have no tested pressure canning for fats like tallow and lard. Without knowing if that is even needed, or what the process should be if it is, I do not recommend subjecting the fat to that kind of heating.  Excessive heating will hasten rancidity changes in fats that cause off flavors. If the fats are rendered and impurities filtered out, there should be no need to apply a canning process.

Extension/USDA does not have canning recommendations for tallow and lard at all. Pure tallow and lard can be kept at room temperature, and should be kept away from oxygen (air).  That is why bottling without canning as you described works. After rendering, and filtering the impurities out, you can pour the warm fat right into jars and allow to cool to room temperature. The best storage for quality reasons is refrigeration or freezing, or at least keep as cool as possible.