CW308,
Given that you run 6 minutes of cooling between shots, the first thing I would recommend is that you take a Lee decapping rod and a small hammer to the range with you (bottom of a Lee Loader works, too) and deprime while the case is still hot. Carbon hardens as it ages. You'll find the warm primer residue falls right out and you can put a dab of something that dissolves carbon, like Gunzilla or Boretech C4 Carbon remover into the primer pocket with a swab and even do the same inside the case. Board member Hummer70 came up with this method, though I think he was using Ed's Red as the carbon solvent. Check it out
in the first post in this thread. Anyway, if you do this, whatever cleaning method you use will work faster and easier at removing carbon subsequently.
For cleaning the quantity you have, rather than set up stainless pins, you might look into ultrasonic cleaning. The pins do put small impact marks on the brass surface. I've not seen an issue with them, but if, like a lot of benchrest shooters, you are trying to squeeze 50 loads out of your carefully selected and prepped rifle brass, then you might want to avoid the extra surface work hardening that comes with those pin impacts.
Below are some corroded .30-06 cases I cleaned in by ultrasonic in a solution of 5% citric acid plus a teaspoon per gallon of dishwashing liquid to act as a wetting agent and suspend brass. Subsequently I tumbled them in Lyman green corncob media and all the pink (where corrosion had preferentially taken zinc from the surface and left copper wash behind) turned to polished brass.
Note that these cases sat in a bag that got flooded, causing the corrosion, and that the primers in them had been their for a good decade, so the carbon was well hardened. I didn't take a photo of the insides of the cases, but I did look with a borescope and found them essentially clean as the proverbial whistle inside. The main point of interest for me was that the inside of the necks be clean for consistent bullet pull friction, and they were.