I have never seen copper coating on austenitic electrodes.
And austenite stabilization is essentially irrelevant to ferritic electrodes.
It is true that austenite 'stabilizers' will either cause austenite to form sooner or last longer (or both) through cooling which can effect microstructural constituents.
But I do not see how the small amount in a rust inhibitor coating would be significant enough to say increase the volume percent of martensite or bainite at room temperature, which would be the issue of concern. And much of it will oxidize and end up in a slag cover anyway, the insolubility of which you speak.