Do you still get any ammonia and nitrite from your latest readings?
Just keep up with the water change. You may do no more than 50~70% partial water change each time to avoid shock the fish. You may also cut back on feeding which can help your situation. Stop feeding the fish for a day or two completely will do no harm at all but dramatically lower the potential ammonia you put into the system. Afterward, you can feed them once every two days with only enough for them to finish within a minute with no leftover. Then once a day with similar amount after everything is back to normal.
Adding aquarium salt in this case would have no real benefit.
If you use Seachem Prime for every water change, you can minimize the ammonia damage to the fish since it can temporarily turn the ammonia into relatively harmless ammonium for up to 48 hours.
I know sometimes the incompatible fish can be peaceful with each other, but all it takes is one incident to make it from "no issue" to "too late". Most Tetra species can be very nippy, and the slow moving betta's long fin makes the perfect target. Just a word of caution. Personally I wouldn't house a betta in an aquarium with anything but bottom feeders. Please don't try to duplicate what the pet stores do. They often keep fish in less ideal conditions for obvious reasons.
By the way, how many Cardinal Tetra do you have?
Your tank is on the small side. Too many fish can also easily cause ammonia spike.
P.S. Use an automatic feeder instead of some "food block" next time you are gone for more than two week.
