OK, so to explore lowering the pH a bit. The safest method is also the natural method, and that is simply by diluting the tap water with non-hard water. Distilled water, RO (reverse osmosis) water, or rainwater will achieve this. Depending where you live, rainwater may be the best and cheapest, and for a 5g tank it won't take much. Collect it out in the open, not runoff from a roof which can pick up toxins.
Use a pail for this, so as not to further stress the cory with more fluctuating water. Mix tap water with rain water until you have the pH you want, and I would aim for around neutral. pH of 7 is un-natural, but 6.8 or 7.2 is what I'm meaning to aim for. Then, let it sit overnight and test pH again. The idea here is that we are reducing the hardness more than the pH, so the pH will not keep rising. Also, the hardness is equally important for the fish. Rainwater normally has a low pH, but some mixing is still important for stability.
Once you have it stable, you can do small water changes over several days until you have the tank where you want it.
Collect some rainwater to use for water changes, same principle. The good news is that once you have the tank stabilized with softer water, the ph will tend to lower a bit on its own, so using a mix of tap/rain water for water changes should work well.
Byron.