If the water is simply sitting in the pail then it will never get oxygenated, there needs to be continuous surface movement.
That statement is not exactly correct. The oxygen-water system will always be driven towards equilibrium. So, even with no surface movement, some oxygen will always enter water, if there is less oxygen than the equilibrium point. The rate at which oxygen will enter the water will be less, possibly lot less, but the rate will definitely not be zero.
Now, the act of pouring water into the bucket, stirring in some dechlorinator, and pouring that water into a tank should be more than enough. Water from city taps should be pretty close to saturated with oxygen. Water treatment plants will sometimes add CO2 to water (which is why you should let the water sit out for a while before testing your taps' pH), but the extra CO2 in the water really does not change the amount of O2 the water can hold. (I cannot say 'has no effect' because that is probably not exactly right, but I would bet it changes the saturation point less than a few hundredths of a percent.)
The only thing I can think of would be if you are on a well system and the well is very very very deep. Even then, if the water coming up is devoid of O2, oxygen would enter the water very quickly. It may not saturate with oxygen really quickly, but I would bet like 80-90% saturation again just by splashing it around. The thing is, it takes very unnatural situations to get the air out of water, and I have never heard of anyone killing their fish with underoxygenated water from their taps.