I was originally thinking you could get a qualitative idea of purity by comparing the reaction rate of the iron and chlorine in the tubes. However, a quanitative method is perhaps called for.
One could perhaps test the purity by liquid volume. After capturing a fixed volume of gas in each tube, carefully seal the tubes. Submerge all three tubes in an dry ice / acetone cooling bath. This will condense the chlorine but leave oxygen in the gas phase. Once the gasses have condensed, mark the height of the liquid on the side of the tube. Remove the tubes outside and unseal them allowing all gasses to disperse. Afterwards, fill the tubes up to the lines with water. Measure the amount of water it takes to fill the line to get the volume of chlorine gas in each tube.
This experiment would require test tubes that can withstand a bit of vaccum pressure. It also operates under the assumption that the catalyzed tubes represent pure forms of their respective gasses. If there were impurities that would also be condensed by the cooling bath, however, they would likely be present in the oxygen tube as well. You could measure the volume of these condensed impurity gasses to get a rough idea of how much to offset your chlorine volume measurements by.
Use density to get the mass of chlorine and then the number of moles. To get the amount of oxygen in the electrolysis tube, simply take the extra volume of the tube and use the gas laws to calculate the number of moles of oxygen in that volume of gas at the temperature of the cooling bath. This process may be involved but I think it would yield fairly definitive results for a given electrolytic cell.