Not sure if I have the correct picture of what you have, but let's see if this fits. If I read it correctly, you are saying your old flange fit inside the 3" coupling and you then broke a piece out of the pipe below that coupling. I can appreciate that you don't want to have to break out more concrete than necessary, but I believe you will have to because flanges don't extend far enough down. The broken pipe needs to be cut off level and a new coupler installed. Then a short length of pipe added to the coupler and the new flange fit on the outside of that piece. Do not even consider the inside fitting flange for 3" pipe. The exact length of that pipe is cut to fit the space, making sure the flange rests on top of the finished floor. You have two fairly difficult jobs to do. First, chipping out the concrete without damaging the pipe. It can be done with hammer and chisel, but easier with a small rotary hammer. The next task is cutting the pipe. Plumbers have the tools to cut a pipe from the inside, piece of cake if you have the tool. Don't know if you can rent one or not. You also might use a Dremel type tool. The remaining pipe will have to be cleaned pretty well to get the new coupler on. The new flange should be the kind that has a stainless steel ring. Avoid the all plastic ones, they can break. Stay tuned, some of the pros with more experience that I have may chime in with some better suggestions.