So I solved the issue. My PSA stock had light milling on the bottom of the part that inserts into the receiver. My assumption is that makes for easier insertion due to the smoothed out flashing but allows vertical wobble with the PSA two-piece tang.
I still have the Stormwerkz insert inbound but I had an Arsenal brand stock that appears almost identical to the PSA except no milling on the bottom.
The Arsenal stock inserted rock solid even without the tang. It is installed now and the only visible difference in the PSA and Arsenal stock is a small "US" raised logo on the neck of the Arsenal stock.
I'd sure like to know which company is the true source of the stocks. PSA, Arsenal, and Atlantic seem to all have the same polymer stock sets molded by a third party company.