So you are taking down a tank, taking out the rocks, and removing the sand, and therefore you are stirring up all sorts of detritus(unless you keep a barebottom tank with a lot of flow). When you set the new tank up, you are running a risk with any more delicate corals that you may have(like sps) due to the dramatic increase of nutrients floating around in the water(caused by stirring up the sand and rocks). It will take your tank a little while to fully return to its former glory, and for things to return to a balanced state. Despite these risks, this is the least risky scenario IMO. Your corals might brown out, but your life support system for your main tank is generally much better then that of a temporary just-set-up housing. Your flow is usually better, the lighting is better, and if it is the summer and hot out, generally your temperature control is better too(better cooling).
The rubbermaid can be done, but imo it is inferior to just putting the corals in the display. There are a lot more variables that you have to watch out for. If you really want to do the rubbermaid route though, you need to use 100% water from your display. You also need to make 100% sure that you are using the right kind of rubbermaid container. There are many different kinds of plastics that will leak all sorts of toxins in to the water, so make sure it isn't one of those plastics. Flow is also important. If you cram a bunch of larger colonies in to a small space, they can block alot of flow from reaching one another. This set up may also be prone to PH swings depending on how your display is set up(my display uses kalkwasser and a reverse lit refugium to keep ph up). Finally, make sure you have a good way to control temperature.
In either of those scenarios, I would just try to sell off some of my larger colonies(and keep frags) to reduce the amount of corals Id have to move. Moving can be a risky time for a tank. Sometimes it works out fine, other times it doesn't. Given that risk, I would rather have some cash in my wallet and less stress on me during the move. Plus, it is fun to be able to trim down what you have, and add some new coral to your mix.