Before blaming the cats, waiting to see if the CPS (or a combo of CPS/cats) is causing the issue. If rings were bad, just warming the car up under no load and then doing a compression test should show that, especially in the affected cylinders that code-out often. Resetting the ECU/battery making it better sure doesn't sound like a mechanical problem going on, or even the cats. I am leaning toward the CPS being at issue.