It is quite possible for me to run the water through the filter faster than the ideal rate but even at full flow the water from the filtered faucet has less chlorine flavor than that from the unfiltered faucet. I usually cut the flow down when filling the brew kettle but still use a campden tablet for good measure.
It is good that you also add the campden tab to the water. You really do have to reduce the flow rate through a carbon filter unit in order to provide complete chlorine compound removal from water. Pushing water through at too high a rate does mean that some of those chlorine compounds make it through into the brewing water. It takes incredibly little of those compounds in your wort to screw up your beer. The campden dose should solve that problem.