Screens only have to be fine enough to stop the brass. Let the pins fall through into a plastic bucket. A piece of hardware cloth bent into a bowl shape will work. A lot of folks just put a plastic collander over a 5 gallon bucket. The pins apparently are not austenitic, so a magnet will get strays (test this with your pins). Others are
using these and just buy their own bucket, but I see they are currently out of stock. Also, currently they are made sized for 3½ gallon buckets instead of the ubiquitous 5 gallon paint buckets.
Thumbler B seems to be what most favor for pins. Shown in
this article. It does rocks, case cleaning, moly-coating (with special plastic liners; moly sticks to their standard Neoprene liner). No difference in the machine needed for that.
There's an old arsenal case cleaning formula (from before they stopped cleaning oxides off brass, so this is WWI era) that's just 5% citric acid powder by weight (add 7 ounces to 1 gallon of water) that works fast and is reusable and good for the ultrasonic. I expect it's a good choice for tumbler, but you may be able to use a lot less considering the time involved, and toss it when you are done. Buying the citric acid as such is a lot cheaper than buying it as Lemishine. 10 lbs for less than $3/lb postage paid,
here.
Citric acid softens water, so you don't really need a lot of detergent. Also, some detergents a alkaline and neutralize part of the acid. Try just a couple of drops of Dawn Essentials (no color or disinfectant or bleach or other additives other than scent) to help cut oil or grease.
Try cleaning the drum friction surfaces with alcohol. Don't use belt dressing, as it lubricates squeaks out and may actually make the drum slip worse. If it still won't grip, mix a little rosin in alcohol and paint it thinly on those surfaces and let it dry.