1. steels and metals suffer by microbial corrosion. But you should note that all Bactria can not have corrosion effects.
2. cooper can have toxic effects again only some batteries. It mean some other batteries can growth on bio film on cooper, without any troubles.
So corrosion problems on cooper depend on Bacteria's type. if some un-corrosive bacteria make a biofilm and develop on the surface, this biofil can not play a bad role against the metal surface, and if some corrosive bacteria (which is resist against cooper) make a biofilm near the metal, then the metal would suffer by corrosion.
All incubators I have seen in the last 20 years have had a stainless inside not copper. I have seen some with copper trays but not a copper container.
My personal opinion without ever looking into heavily is that it will not make a difference in your contamination rates. Most problems with contamination arise from poor technique or poor maintenance of the incubator. The best solution I have seen is with the ability of the new incubators to run a sterile cycle which sterilizes the whole inside including all metal seams. I once worked in a GMP environment that was regulated by the FDA. If there was a significant benefit, I am sure the sales people of the equipment would of let us know. (empirical ).