Is there any way to quantitatively find when a reaction has completed?
Option 1: Spectrometry

Easiest method, in most cases accurate enough. If you know that there are no side-reactions, this is a safe bet. Simply measure some characteristic wavelength for one reactant (preferably $\ce{Fe^{3+}}$) and one product (I would go for the $\ce{Cu^{+}}$ there) and then make measurements of your reaction mixture on the fly.

Option 2: Titration

Not that recommendable if you need any reliable results, but if you are low on budget and/or want a didactic effect. Take a little sample from a big flask and do a $\ce{Fe^{2+/3+}}$-redox-titration.

Option 3: Electrolysis

Don't know how to implement that, but basically you would take the sample and measure the amount of electricity needed to convert all $\ce{Fe^{2+}}$ in that sample back to $\ce{Fe^{3+}}$. Probably more accurate than spectrometry as the measurements of currents can be done quite precisely

You could also do some weird stuff like quantitative chlorine-NMR, but I can not think of any benefit that you would have with that.