I have also thought to eliminate the aluminum from the cooling system. However, all components will then have the decimal point in their price tags moved one place to the right. In addition finding a radiator of suitable size is a little bit hard. There are some suitable radiators which are intended for car air conditioning (heating element) but those have usually very strange shape and are not widely available.
It is my intention to make this project doable for everyone so the radiator has to be affordable and available.
So, how about eliminating the copper ?
The conductivity of aluminum is about 60% of conductivity of copper. With these frequencies the
skin effect plays a minor role on current distribution in the coil so making the coil of aluminum with thicker wall might just work. Bending the aluminum tube would also be easier if the tube wall is thicker.
With aluminum the challenge is to get the connections reliable. It is possible to solder aluminium with correct flux but one has to be really careful with temperatures and the vapors of that flux are nasty.
On use aluminum would behave better in the coil as it would not corrode because of the oxide layer. Copper will get quite nasty look after some use even if it is cooled.
One more option would be to plate the copper tube inside with some suitable metal. The trick here is to get the coating complete so that every bit of the inside of the tube is covered. Here a suitable metal would be zinc. Tin might just do it and further more, the tin can be applied chemically without electrolysis or excessive heat. Still this method will be nothing but easy as the tube is rather long and slender.
Last but not least machining copper and achieving an acceptable surface quality is something I have found to be really hard to do. So probably the heat sinks would need to be made out of brass instead of copper.
Edit:
Introduced Current aka Cathodic corrosion prevention
Actually the ion filter is far simpler than I originally thought. In this case it will be an aluminum tube connected to positive voltage through a suitable resistor. The resistor limits the current to a suitable value so that the sacrificial aluminum tube corrodes at acceptable speed. If this tube is thick enough (5 mm wall thickness) it will practically last forever. Further more it is easily replaceable.
So how does this work. Instead of letting the copper ions to escape from the coil, the positively charged sacrificial anode forces them back into the copper surface. The sacrificial anode will release aluminum ions into the cooling system which are not harmful for the aluminum parts. The voltage applied on the sacrificial anode could be 5 volts (readily available) which should be high enough to keep the copper ions where they belong to.
The aluminum tube(s) are placed between the copper coil and the rest of the cooling system. The radiator and other aluminum cooling part still need to be isolated in order to minimize corrosion when the device is powered off.