Attach about a 1" thick piece of scrapwood to faceplate - you can use screws, but not drywall screws - they will break.
Turn round, so it looks pretty and doesn't hurt you with corners that stick out.
While it rotates, mark center. Drill hole through with hand drill (stay straight as possible.) VERY LOW SPEED on drill, also have lathe running slowly, will assist you in staying straight. No vibration, you're straight.
Now, insert a dowel (smaller diameter than the hole you will ultimately have to drill in your bottlestopper - some mechanisms have different hole sizes).
Once you get it so you can run your lathe and the dowel is spinning "true", glue the dowel in the wasteblock on your faceplate. Now, make a hole in the material you want to use for the stopper (same size as the dowel) and turn your stopper. As always, use tailstock as long as possible, but the faceplate won't slip - you CAN break the dowel, but you have more dowel stock and more waste stock, start over if you can't just drill out the dowel and use a new one.
IF you get slippage, insert toilet paper (You keep a roll of this near your lathe at all times, don't you?) as a spacer to "snug up".
Try it, it worked for me until I got a scroll chuck.
(Idea courtesy of Dick Sing - this particular application is mine)
Good Luck, have fun, be careful in drilling - you CAN get hurt if you don't pay attention.