BB, If you have never shaped a cork grip before, I recommend turning the cork on a mandrel (a dowel or piece of threaded rod) in the drill press. Once the rod grip is shaped, slip it off the mandrel and glue it to your rod. If something 'catches' on a drill press, your rod blank will be shards before you can blink.
Also, write an outline of the steps you will follow in order. For instance, if you shape the grip as you describe, how will you slide the reelseat onto the rod butt. From the open end? Or is the interior of the reelseat tapered to be mounted on the blank before gluing up the grip?
I've posted this photo before. I use a home built rod lathe for turning grips, wrapping thread, and applying and turning wrap finish. When I turn grips, I have a box fan with a furnace filter behind the lathe to pick up most of the dust. Otherwise the dust takes a week to settle enough to wrap guides or apply finish. Better still, take the lathe outside to turn the grip. I built the lathe from Big Box store dimensioned lumber with a jig saw and a cheap back saw/miter box. No need to move the vehicle or plan an expedition to find the table saw. I'll take a few more photos and make a drawing in a day or two.
Note: save a baggie of cork dust. Mix it with wood glue for filling mistakes.