I'm wondering if I'm allowed to run on an already completed save? (I'd start at C1 and run as normal but the save would be counted as complete)

Mostly asking since I'd be able to skip the shooting minigame in C4 if this is allowed

You must start on a new game file which is very important. We use IGT for the full game run and that only starts at 0 when you start a new game

If you are asking for the chapter 4 il the answer is still the same just to make sure there is no difference between IL and full game run