Rob,
You might want to read through this thread from "that other site."
https://www.benzworld.org/threads/2004-e320-oil-consumption.1694417/#post-5620892
If you read through to the end, you'll see that OP had eventual success with the "procedure" discussed in the thread and switching to a 5W-40 oil (often the independent shops fill with 0W-40). I'll note that the OP in the thread had the M112 V-6 as opposed to the M272 V-6 in yours, but there have been other posts over the years on that site discussing using the same procedure and switch to 5W-40 for the M272. I was too lazy to search in more detail, honestly.
Also, I'm sure you know this already, but these motors can lose a lot of oil due to leaking cam plugs and a leaking crankcase breather/oil separator.
Anyway, I'm not saying the procedure discussed in the thread will work for you, and I've no experience trying it, but it seems to have helped some folks. I haven't had the oil consumption issue with my E350 wagon with the M272; it has about 150,000 miles on it now. I've never followed the FSS notifications since I bought it in 2008; 12,000 miles/1-year seems like an awful long time to me. I just do the oil and filter every 6 months, regardless of mileage. Yes, I'm probably wasting money doing so, but it's an easy schedule to stick with.
I hope you're able to solve the issue without having to spend a lot or replace the car. Best of luck.
-Chris