How do you experimentally calculate the ionization constant of the carbonate ion?
So far, so good. You'll have to graph the $\mathrm{pH}$ of your system with the volume of titrant you've used to construct what's called a titration curve.

When you do the titration curve for this experiment, depending on the relative strengths of the bases involved, you will see one or two points of inflection points that will signal that the equilibrium conditions have changed. This will tell you if you need to take into account the second ionization. If there is a second inflection point, that's the second ionization.

Without actually going ahead and looking at the $\mathrm{p}K_\mathrm{a}$ of $\ce{H2CO3}$, I think for most computations, it could be omitted, since it is a second ionization which has a much lower constant than the first. Not unless you're looking for zwitterionic states or something like that.