I'm pretty sure that Processing is incapable of doing this.
Another thing to note is that you shouldn't call size() with variables.

The workaround method uses PGraphics.
PGraphics are just like the main canvas except you aren't drawing to the screen, and they can still be saved.

On my computer the sketch is indeed of width 2200, even though the window is only as big as my screen. When I save it, the result is in the expected (w,h) resolution. Alternatively you could always use PGraphics. That's probably a neater solution anyway, because you can scroll over the big canvas like a map even if your sketch window is smaller.

Test Code

1. void setup() {
2.   int w=2200;
3.   int h=1000;
4.   size(w, h);
5.   background(255);
6.   print("size("+w+","+h+") >> size("+width+","+height+")"); // 2200x1000 & 2200x1000
   
7.   for (int i=0; i<width; i++) {
8.     stroke(i%255);
9.     line(i, 0, i, height);
10.   }
11.   save("save.png"); // also 2200x1000
12.   exit();
13. }

Running your code with exit() removed, the image is correctly saved with the full size, but the visible window is still too narrow. Strange.