DIY galvanize copper on stainless steel?
It's apparently easy enough to plate a layer of copper on stainless steel.  The problem is getting a layer thick enough to be visible, and getting it to adhere well.  @fred_dot_u's answer beat me to the discussion in the linked article.  For results suitable for typical industrial use, it generally requires chemicals you might not want to mess with at home.  Even manufacturers farm it out to companies that specialize in it.  

That said, what you can reasonably do at home might be adequate for what you need.  Here's an Instructable that describes the typical way to electroplate metals with copper at home.  Stainless steel will be more of a challenge, and the plating may not be durable, but it would be a relatively easy thing to test.

There's also been some work with a gel that you apply to stainless steel and then abrade the surface under the gel to expose metal that is instantly plated.  It immediately forms a layer of copper that's just thick enough to color the surface, and then the process stops.  This video shows the process.  

It links to this video that describes the gel and shows it being used in an additional way.  They use some copper wire connected to the stainless steel and embedded in the gel.  The gel acts like an electrolyte, and the wire and stainless steel create a battery that deposits a thicker layer of copper.  Unfortunately, I didn't notice a specific description of how to make the gel, but you may be able to find it by exploring their links.  Just knowing that the stuff exists means it's possible to do it (although they do it in a lab, not at home).