You pose an interesting question there, it all depends on the expected loads that would be applied. For instance is the winch intended to be used in anger or just for display purposes, If not spring steel then certainly a decent grade of steel would be needed not just flat black bar, bearing in mind that the band will need to be rolled into shape and drilled for end fixings and rivet holes.
Have you by any chance checked to see if the original winch manufacturer is still around? Alternatively you could try talking to the company linked here.
http://www.industrialfriction.com/brake-bands/