If you are going to say that la nina can be responsible for cooling earth's climate system during a given year, you need to be able to explain where in earth's climate system la nina is hiding the heat. Or, if those joules no longer remain in the system, how were they destroyed?
Because the charts answer the question, and if you considered them you might realize what the answer is, without accusing me of believing in "insane stuff" or the "La Nina fallacy". The charts diagram the characteristics of La Nina and El Nino. Are they sufficient to allow you to understand how they work? If not, do you have any particular specific questions?
you need to be able to explain where in earth's climate system la nina is hiding the heat.
It's not hiding. Look at the La Nina diagram and examine the thermocline depth in the western Pacific. (Compare to the "normal" condition to make this clearer.)