Test kits that measure KH, work on the principle of adding acid to the sample, with the amount of acid added (number of drops) determining the concentration of things buffering against this added acid.
When CO3 has formed H2CO3, it can no longer accept anymore H+ ions, and thus, it cannot buffer pH against the addition of more H+ ions, and thus, those dinky KH test kits won't register the level of CO3.
Think of pure water that has no KH. pH only changes if you change the ratio of H+ to OH- ions. Just because the water doesn't have KH, doesn't mean the pH just randomly bounces all over the place, however, since there is nothing in the water for the addition of H+ to bond to, any H+ ions added to the water have a direct effect on pH. If you raise the KH of this pure water to 10 German degrees, and add a bunch of H+ ions (pH down for the win) to the water, the pH will drop with the addition of H+, and then rise to the original pH again. We all know the affect, pH down stuff gets asked every other week. The reduction in pH from pH down is the addition of free H+ ions to the water, and the subsequent rise in pH is the bonding of those free H+ ions to CO3 and/or HCO3, and thus reducing the concentration of free H+ ions affecting pH.
So again, since H2CO3 can not accept free H+ ions, it's affect on KH = 0.
The reaction is indeed reversible. However, one must not forget the final reaction, CO2 + H2O <> H2CO3. The atoms on the left hand side of the equation equal the atoms on the right hand side of the equation. One Carbon atom, two Hydrogen atoms and three Oxygen atoms.
As H2CO3 concentration is increased, some of this transforms into CO2 + H2O. Since the concentration of CO2 in water always wants to maintain equilibrium with the atmosphere, as the concentration of H2CO3 is increased, and hence the concentration of CO2 in the water is increased, some of this CO2 is lost to the atmosphere. In other words, one Carbon atom and two Oxygen atoms are lost to the atmosphere, quite a large chunk of CO3 don't you think. In fact, all that remains is two Hydrogen atoms and one Oxygen atom (H2O). In this situation, the addition of free H+ ions has had a direct effect on the chemical species (H2CO3, HCO3, CO3). Where we inject CO2 into the water, the existing HCO3 and CO3 species remain unaffected, as the increase in H2CO3 is thanks to CO2 + H2O, rather then the transformation of the existing HCO3 and CO3 species through the sole addition of H+.
edit: Also, I don't think KH test kits measure alkalinty, I think they do indeed only measure KH. Where they fail is when other sources are present in the water that affect pH. Since these test kits titrate the sample to a pH endpoint, any fluctuation in pH from sources other then KH will affect the reading. But it's just a theory at this stage.