I had the exact same problem. What I did was switch over to a 12" rough toilet.The waste line had to be cut off but I left about a 6" stub(I used an angle grinder to cut through the cast iron stack pipe) I coupled a piece of 3"PVC to the stub I left on the stack pipe using a hubless connector that I bought at Home Depot. You do have another problem in the fact that the hole in your floor is too big for the toilet flange. You will have to make the hole even bigger now and put in a larger piece of wood then cut a smaller hole for the flange. The easiest way to attach the new piece of floor would be to nail a 2X4 to the floor joist flush with the underside of the floor to give you a nailing surface instead of trying to cut the existing floor back 3/4 of an inch on the top of the floor joist. Before attaching the new piece of floor carefully cut the hole in it for the new flange. To get the hole right run the pipe from the new closet bend up through existing hole in the floor. Measure then cut the hole in the new piece. Glue the pipe in place then put the floor down then cut the pipe off at the floor level. Be sure when you cut the hole in the floor that you cut it to the size of the floor flange and not the 3" pipe! Good luck.
I tried to attach a picture but the size was too big so I couldn't download it.