You could go back to the shop that originally polished it and ask them to set you up with some sort of daily/weekly protocol for maintaining it.
One widely sold brand with a good reputation is 3M Imperial Wetordry, which is available in a full range of grits up to 2000. (Despite the name, grits above about 400 should always be used with water - they clog rapidly when used dry.)
2000-grit used on plexiglas yields a finish that, optically, is almost as good as glass. I doubt you’d need anything better for copper, and even 1000-grit is probably sufficient. The first time you polish the surface, you’d need to start fairly coarse and work steadily up through the grits.
Is it wrong to assume you are working for or in conjunction with a major university? If not, check to see if they have a Materials Engineering department. Those guys routinely polish metal samples to be both flat and free from visible scratches (this is necessary for microscopic examination, an Optical Microscope has a very shallow depth of focus at 500X and scratches disturb the metal which compromise the examination). If you don’t work for a university with a Materials Engineering dept, there is likely one close to you that could help.,
As for maintaining the surface, they may also be able to suggest a solution. I would think that a electroplating of gold would do. The thermal conductivity of gold is close to that of copper, and with a thin coating, I doubt it would make any difference.