Your mention of "Extreme" reminded me there's a "European Extreme" difficulty level, I only made it halfway through MGS3 on that (attempting no kills at the same time)
The only strategy that somewhat worked for me was perfect accuracy tranquilizer shots, to knock them out instantly. That's probably the hardest game mode I've ever played.
I also completed MGS3 on euro extreme, and was about an hour from the end of MGS2 on euro extreme (the action sequence right before the MG Ray fight). I was playing the PC port, and let me tell you: aiming the automatic weapons without pressure sensitive buttons is nearly impossible. I gave up eventually and decided that my prior run on Extreme had earned me enough gamer cred. Finishing euro extreme wasn’t worth it.
On the other hand, I loved MGS3 on euro extreme! It really required mastering every trick in the game. Every little advantage you could squeeze into a boss fight was essential. Escape from Groznygrad was hell, though. By far the single hardest part of the game.
The only strategy that somewhat worked for me was perfect accuracy tranquilizer shots, to knock them out instantly. That's probably the hardest game mode I've ever played.