Once I stopped trying to get the clay true to the bat, all of my centering issues went away. I keep my hands off the bats. I came to this realization after watching people on youtube thow beautiful big vessels on bats that were stuck to the wheel head with clay wads. The bats were moving up and down and roughly centered. It has no effect because they were throwing in relation to the center of mass of the clay and not trying to true it to the surface of the bat. Try to open in a 'V' rather than a straight wall. The V from the edge down to the center is self-centering all you need to do is control the lip with a thumb or finger of the top.
This guy throws with a wobbly bat:
This is also a good clip:
I only have YouTube and this site to learn from, there are no facilities or people in Central North Dakota to teach. Hope it helps or at least gives you somehting to try.
Try doing that with a 25" cylinder-it doesn't work!