Your results show some consistencies, but there are other values that seem "odd." It makes sense that the pH would change, but just adding lime should not drop all your other ions. The reaction, for example, has little to do with Cl or SO4, but you see the results change. This makes me question a few things. I would question the following :
a) how many samples did the Ward Labs test?
b) the Clark process only affects carbonate. However, adding sodium carbonate (or creating substantial side reactions) could start affecting Na and Mg levels. I would be surprised at the levels you have measured of Na, however, just from a lime process.
You may want to investigate, what temperature you did things at and how much mixing was involved, the time length for reaction, etc. If all else fails, you may want to talk to the manufacturer and get a "tech" chemist on the phone.
Also note that commercial quicklime has other impurities (silica, iron, alumina, etc). If adding lime, I would want to make sure it is a pretty high purity grade source..For example, I wouldn't just use the stuff from the garden center....otherwise you may be changing alkalinity, but adding other things you really don't want to ingest. You may want to look at whether these are in your water before you use it. My suggestion whenever testing water is to look for results from three samples.
Best of luck and brewing to you!