Have you tried grassfed meats? I wonder if the grassfed would have the same effect on your skin as grain fed.
Are you taking any supplements? If not, you might want to research them and figure out what to take to get that systemic inflammation under control. Until you get that sorted, I would imagine you're just in a negative feedback cycle. (the ones that seem to help me most are fish oil/krill oil, vitamin D and B vitamins, but your mileage will most certainly vary as I've concluded that everyone needs their own "cocktail" and the only way to figure that out is to start playing around with the supplements.)
Until then, I'd get protein from fish as much as possible (yeah, you'll get sick to death of tuna, I'm sure, but I'd hazard a guess you might be able to tolerate some of the other stuff once you get the inflammation sorted.)
If I had to choose, I'd take grassfed/organic meat over organic veggies. And I didn't see where you live, but local produce is a good bet.