Question: Is there or is there not tile and concrete under the screw holes on the existing flange? It looks like it's mounted on top of the finished floor, like it's supposed to be, but just isn't secured through the finished floor to the subfloor like it's supposed to be. I.e. is there a reason that you can't just just bolt it down through the tile into the concrete?
Or am I mistaken and the hole around the pipe actually is so wide that it comes outside the screw holes on the flange? If that's the case, then I don't think you'd need to chisel to get an outside-fit on there.
But, assuming that I am continuing to miss stuff (as I was earlier), the simple answer is, "Yes, if you need more room you can chisel."