Yes, a cycle is needed, even if you transfer the water from your display. The water does not hold any biological bacteria. Biological bacteria is found on surfaces, such as substrate, live rock, bio balls, sponges, bio wheels, etc. When you start the QT, biological bacteria needs to form in order to convert ammonia to nitrite to nitrate. What Turbo Start does is it rapidly adds massive amounts of live bacteria into the water to attach onto your filter.
I personally would not copper any of those sensitive fish (clown excluded). All of them are copper sensitive and may respond quite negatively to the copper treatments.
I myself would do a hyposalinity treatment on them instead. If you are curious as to how to do one of those, check out the step-by-step instructions posted by Beth at the top of this forum.