I don't think all big rockets are negatively affected by SAS. My Space Shuttle stands at almost 450 tons (475 with maximum payload), with lots of struts connecting the solid boosters, orbiter, and external tank, and does very well with SAS. At first, (when I was very noob), I did not use struts, and the solid boosters wobbled, and the whole thing fell apart. With struts, and SAS, it is very smooth. You've just got to strut it more (and at the right places).
By right, SAS is supposed to stabilize by exerting opposite forces to the current direction you're rotating towards. The reason why it does not do so on larger rockets is because they "bend" too much. When a rocket "bends" (oscillates) too much, by the time SAS corrects the pitch/yaw/roll of your rocket, your rocket will have oscillated to the "other" side, making it even more unstable. In other words, your rocket "bends" faster than SAS can correct each oscillation. That is the reason why "bendy" rockets tend to break when using SAS. If you find it hard to control your rocket, try making your rocket less "bendy" (make it more rigid). Also try to add control surfaces or more vectoring engines and see if it helps.