How to bore out step pulley

Another idea would would be a steady rest on the big end if it is turned true to the bore, then you wouldn't need as much grip on the small end.
 
Wrap a roller chain in the pulley grove and clamp the jaws on the chain.
 
I would set it up in a 3-jaw clamped to my mill table, small end down. Dial it in and ream 7/8. If there is any doubt about the end face being flat & perpendicular to the bore make a 22mm mandrel and face the pulley in the lathe.
 
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