Painting will be ok, it don’t have any effect on convection, and should improve radiation a lot, where copper is weak. The best color for painting is silver; it can emit radiation but don’t absorb anything because it’s reflective capability. Second to it comes white; the worst is black because it absorbs anything received.
That container you speak seems good; I guess it’s heavy being pure thick copper. It could be easy to improve its natural convection capability by soldering vertical copper fins to it.
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