How to cut center stile from cabinet?
Definitely cut the stile so as to leave a bit of wood behind on the cabinet. Then using a sharp blade and a straight edge, score the veneer. Many shallow passes are better than few deep ones. If you can cut all the way through, you can probably remove the veneer before you attempt to sand down the stub.

And if you are careful to not muck up the stile too much then you can attach it to one of the cabinet doors.