Do I need to include calcium chloride in this reaction equation?
From an instructor's point of view, you have accounted for the reaction that has taken place during titration.  The point you are making surrounds the conditions the titration took place under (as you pointed out, solubility is altered, for instance), but not the chemistry of the reaction, in and of itself, which you are focused on.

In essence, you are doing the first titration in aqueous calcium chloride as well, because as soon as the titration begins, what is formed?

That said, I feel I may be missing something more of the other potential rxn conditions or possible expectations of your evaluator.