>I know you typically need at least a 3" pipe for sewage

Just a little context on this statement, i.e. "at least" often a 4" pipe is problematic because it's too big and shit gets stuck to the sides and there isn't enough water flow with newer efficient toilets to fill the pipe and flush it out. So 3" is like the happy medium, rather than a bare minimum.

I have no idea about using a 2" pipe for this purpose so can't weigh in on that. What does the code say?