Since this forum is externally sourced I do not know too much about the coding of it so I am unsure if HTML tags can be embedded. I will run some test on that later to see, but for now I will focus on your other questions.
My thoughts regarding Lithium battery capacity at high current draw rates are that the high draw capacity would be specific to the brand of battery, and to try to obtain specifications from the battery's distributor regarding its capacity at a certain continuous Amp draw. Without these specifications it would be a guessing game of how they would perform under a certain load. I would not expect these type of specifications to be available however it never hurts to ask anyways.
All battery chemistries that I know of have a lower capacity as the current draw increases. That just seems to be the nature of batteries. I know some battery chemistries are better at this than others though.
Our selection of Lithium batteries is limited because we are wary of unproven brands that may not be trustworthy. We only sell Lithium batteries that are distributed by companies based in the USA who will take responsibility for them if they malfunction and catch fire. If you are concerned about the safety of Lithium batteries then we recommend to only purchase them from companies based in the USA who have been in business for several years and are well established. If Lithium batteries are purchased from outside of the United States then the company that sold them has no liability to repair or replace property that was damaged by the batteries if they malfunction.
It sounds like you are well aware that the maximum Amp rating of the controller needs to be the same or less than the maximum continuous discharge Amp rating of the Lithium battery, However I thought I should mention that in case anyone else reading this does not know.
Please let us know if you have any questions.