I would change the water twice a week in the small tank. otherwise ammonia levels will spike as well as other nitrates. I had to do an emergency water change in my 5 gallon. the levels were bad . and the betta feels allot better now. I have a filter in there. but water changes are needed in a filtered tank also.
Cloudy water almost always means ammonia which is BAD. I'd personally be doing a full water change every other day in a 1 gallon. If you only ever do partials ammonia will still build up - trust me I had it happen in a few 4 gallon QT tanks and I couldn't believe it because I was doing partials and it had only been a week and a few days. It took around 4 90% water changes to get the levels down to 0. Generally once it's cloudy ammonia levels are quite high. Bacterial blooms are more rare, and in this situation I wouldn't think that is the case. They can make the water smell quite bad.
Luckily the quick and easy fix is to do more frequent water changes. I'd do a 100% water change every other day. As long as the temperature matches, the water is from the same source, and it is conditioned this will not bother your betta and in fact will make him much healthier and happier. If you could upgrade the tank size your betta would love it and you would have to do less frequent water changes.
I have Cycle and am kind of on the fence about it. It's definitely not a cure all, but it does "seem" to help cycle a tank faster and also "seemed" to help when my 29 gallon when through a mini-cycle. There's no way to tell for sure thought if the same effect would have happened if I had not added it.
i sort of had the same question. Bluey's tank looks really clean and Crumb's was really cloudy. That's why I switched to sand. I can now see everytime he poops (fun!) and i can get out the trusty turkey baster. I feel confident he is much better off this way. :) And i feel like i'm interacting more with my fish. Sometimes i wonder what they're thinking when they just wiggle and stare... wiggle and stare....