To answer your question, no, I do not think online courses will help. You can self-study for free if you are motivated enough to do so. Many kids do that, including my own. An online course is effectively self study, but you pay for it. And, I will take the liberty of quoting @skieurope 's oft-quoted statement from AO’s at Columbia, et al: “Please don’t do that, it doesn’t impress us.”
It will not be negative against you. It will not help you though.
I am not sure why you think you have an easier shot at Columbia than Harvard. The acceptance rate is 6%. You have no better shot there than at Harvard, for all intents and purposes. By all means, apply, but Columbia and Cornell reject thousands of perfect applicants every year. If your school’s profile states that stduents are allowed only two APs , then you will not have that held against you. Appeal the rule, maybe that will work. And talk to,your GC.