I've had nickel wear off cases just from resizing, long before the stainless pins.
Nick,
A couple of things come to mind: One has been covered: The pins need to be tumbled by themselves for awhile when new, not only to remove lube used to help the process of cutting them, but to dull the edges at their ends where they were cut.
The main issue seems likely to be a galvanic reaction of some kind. That could happen in either acid or base solutions, just as you can make either acid or alkaline batteries. You probably aren't getting the exact same pH in your solutions every time unless you mix components carefully. There is probably an optimal pH range you can operate in and that you have to adjust the mix to achieve.
What you could try is a pH neutral cleaning solution. Citric acid is commonly used to adjust detergent pH in shampoos down from alkaline, and you could do that using one of the inexpensive plant pH meters from a big box garden center.
Amazon has one for under $5. Just keep the electrodes clean with a non-metallic (don't want to contaminate the electrodes) cleaning pad, like a Scotch-Brite pad. These meters work by creating a battery by galvanic reaction between their two probe materials in an aqueous solution. They don't need batteries, for that reason. So if you neutralize the solution well enough that it doesn't show on the meter, I don't expect you have enough ionization to cause trouble for your nickel cases, either. You can, however, check the solution with the meter part way through the cleaning process just to be sure you aren't changing the pH by getting small amounts of metal into solution.
You could also just keep a record of pH readings for your solutions and note what they were when the solutions seemed to behave well, then aim adjustment at those numbers.
It is possible fine particles of brass are embedded in the drum surface that could react. You might try tumbling some water and vinegar with the pins for an hour or two to be sure to clear the surface off, then rinse the pins off and tumble the pins in water only for a while to clean them, too.
You could try using distilled water or RO DI water as a base mix, just in case traces of something in your tap water are contributing to the issue.