Laser engraving copper is routinely done using fibre lasers, so I don't see there would be any major issues, but you would have no doubt have to play around to get it to work in your application, as you routinely do with these sorts of things. Reading it, as you have found out, could be an issue, smart phones are often used to read codes, and many have good quality cameras on board. Some have the ability to operate in a macro mode, which is probably what you would need. It may mean writing, or re-writing code to be able to do that.
If the wire oxidizes, or the bar code generation over heats the wire, then the contrast between the engraved on non engraved portions of the wire may be low, so further work may be needed to process the image as well, again, it will be a case of playing around until you get a satisfactory outcome. Is it possible to stamp the wire where the code is placed to flatten it out slightly, which may improve the writing and reading of the code. Best I can offer in the way of suggestions without knowing more.
Cheers,
Richard