I must configure CKEditor to add a class-attribute to every p-tag in the content. You can do something similar with config.format_p but it will only apply the class-attribut
I had the exact same problem, and after a while of messing around I finally found a one-line solution:
config.format_p = { element : 'p', attributes : { 'class' : 'yourClassName' } };
All you need to do is put this code in config.js and it will work :)
config.js