The problem with moving all the fish into another tank is that the water will quickly become polluted. There are bacteria in all three of the filters and on every surface in the tank - the glass, the decor and even the gravel to a depth of a cm or two. Moving the fish and just one filter will mean not enough bacteria going with the fish so you will have to monitor the ammonia level very closely. Moving all the decor you can cram in the smaller tank would be of some help.
However, keeping a very close eye on the ammonia level in the small tank, and doing a water change if necessary, means that it is do-able. And yes, keep the left over filters in a bucket of water, together with any decor that won't fit in the smaller tank. The bacteria won't starve over 24 hours, but they do need to be kept wet. Use some water from the big tank in the bucket. It would be best if you can turn the filters on in the bucket so the water still flows through the media. If this is not possible, take the media out of the filter and let it float in the water. The bacteria may not starve in the time the filters are in the bucket but they could suffocate with no water going through the media.
And transfer some big tank water into the small tank along with the fish, then if it is a water quality problem the fish won't be shocked by putting them in completely new water. If you put the old tank water back in the big tank after the clean, just topping up with new, it won't affect the fish too badly.
If you could manage to do the clean in one day, and get the tank set back up again within 12 hours, that would be even better

The clean will kill some bacteria - those on the glass and the gravel. There should be plenty left in the filters and on the decor to make up the loss quickly, but still monitor both ammonia and nitrite for a few days. If you don't feed the fish for 1 or 2 days before the clean (unless you were planning on doing it today

) that means they would make less ammonia while in the small tank. And feed very lightly for a few days after the clean.
As Cod says, most diseases are down to poor water quality. Poor water quality stresses the fish and stressed fish have lower immunity so they come down with things easier. You mentioned whitespot - do the fish look as though they have been sprinkled with salt? Once the fish are back in the big tank, watch them for signs of disease.
Can I ask what your maintenance regime has been? Eg, how often do you do a water change, how much do you change, do you clean the gravel when you do a water change. Do you use a water conditioner (dechlorinator) when you do a water change.
What kind of tester do you use, liquid reagent or strips?
Do you routinely add anything to the tank other than dechlorinator?
How much do you feed the fish? The usual recommendation is to feed no more than the fish can eat in 2 minutes.
Anything you can think of to tell us will help in tracking down the problem
