Not sure on cost, but a simple way to warm the copper would be to pass hot water from the hot water lines through it. Actually, to get the temperature right, you might have to have something (like an anti-scald valve) to mix hot and cold water together so the pipes don't become too hot.
On idea would be to build the rack out of standard copper tubing with sweated joints with a water input and water output. The water input would come from the mix of hot/cold water. Then, rather than wasting the water, connect the output back to the hot water tank. Basically, setup a hot water recirculating system. Then to turn the rack "on", have a recirculation pump connected to a switch (perhaps a timer switch). So you simply turn the timer switch on for say 15 minutes, in about 5 minutes, the pipes should be tosty warm from the hot/cold water mix moving through it, your towels are nice and warm when you step out of the shower, you haven't wasted any water, and the circulation pump automaticaly shuts down in 15 minutes to avoid wasting more electricity running the pump all day.