Seems to be some confusion on the question. first step might be to find out WHICH your tap water is treated with as it would be really unusual to have treated the water with both. Unless you are on a small private well, etc. there should be a water confidence report to let you look at what treatment is used.
That really can be step one to looking at alternate methods of getting the treatment used out of the water as chlorine can just be stored for a bit and let it gas off, while chloramine is designed to be more persistent, allowing the company to use less and still get the correct residual to the far points in the system.
But, if we assume you are not trying to avoid using Prime or some other treatment product, the treated water being stored is pretty much the same. I would not favor a totally sealed storage as water without movement tends to get stagnant and all the remaining minerals can do weird things if the water is not stirred a bit like with an air bubbler, powerhead, etc. Just better to not let it go that way, even if the result isn't something we can see or test. Scum can form on the top if not moving and that can lead to water with low O2 content. Kind of like leaving a glass of water setting out overnight tastes a bit "odd"?
Can you see a way to keep the water in the bucket but also keep it stirred? That is an option I use but on a much bigger bucket! When reaching a top of 13 tanks, I wanted lots of good clean water at the right temp for any emergency that came along or when I wanted to drain a tank down for catching fish to sell. I do have chloramine in the tap water, so I did continue to treat with a dechlor product but letting it set with a heater in a big barrel was great for times when I needed it. Keeping a powerhead to move the water to the surface to get the good stuff in and the bad gas out was fine for me.