Tom, I use an ultrasonic cleaner and vibratory tumbler in combination with each other. Even though the overwhelming consensus is that pistol brass primer pockets don't need to be cleaned, that doesn't work for me. I saved up my pennies for a Hornady Magnum sonic cleaner that will handle well over 200 45ACP or 300 9MM cases in a single cleaning. Smaller would be OK, but my weekly volume is anywhere from 300 to 500 rounds per week. Run a continuous 30 minute cycle, rinse, pat dry with a towel, toss into corn cob tumbler for an hour. What comes out is like new shiney brass inside and out in A LOT less time than just tumbling in media or using the stainless steel pins. Overkill, probably, but I like the results.
Consider combining the processes you mentioned in your initial post. Use a solution of Dawn and Lemishine to soak the cases for several minutes with some agitation, rinse, pat dry with a towel, and tumble for an hour. Add some NuFinish (break up the clumps before adding brass) to the media and you'll have decent results inside the cases.