As I understand, a lithium battery uses chemical reaction to generate electricity, as such, increasing the contact area between the electrode and the solution will increase the production rate of ions within the solution, but this may reach a maximum point where the solution is saturated or depleted that the area would have little influence on the reactions, but I would say, with some though not absolute certainty, that increasing the area of the electrode would speed up the discharge process.
Oh, by the way, I am not sure how good this site is for practical matters like this, most people I came across so far seem too involved in their abstract business to give useful answers, rather they just point out little things in my post that they found unappealing. Anyways, I hoped my post helped you in any ways.
