I've been lapping scope rings for fifteen years or so, ever since I ran across a SAKO which had scope problems due to the scope being bent by the rings. I went searching with a micrometer for something appropriate and found a 304 stainless tube about one thou or so under which works really well. (Most scopes are under - they vary quite a bit so check your brand.) I bought a kit assortment of lapping grits from Brownell's but you could buy the same much cheaper at an auto parts or machine tools store. I use 320 grit to do the cutting and then finish off with 600 so as to not scrape up the scope. I twist the lap as I reciprocate it giving a crossed helix look to the lap marks. You want to renew the grit ("charge the lap") fairly frequently to maintain the cutting action. The whole job will take you maybe ten minutes. I don't mess with the cap as it seems to find its place OK.
Of course this means you can't use those pretty Conetrol or Buehler rings. On the other hand, they seem compliant enough that scope bending isn't a problem.
Materials appropriate for the lap are things which are fairly soft so they will hold the grit - the grit becomes embedded in them. Copper is traditional but it wears quickly so I like the soft stainless better. Usually copper is used in a split lap which can be opened up to compensate for wear. Not appropriate here. I've used split copper laps to finish Ruger revolver chambers, for instance, and gotten finishes as good as those in S&W chambers without an appreciable enlargement of chamber diameter. (Hey, with those huge tapered holes, what's another thousandth?)