The steady rest eliminates the wobble. Here’s my setup.
The blue spiral tubing is an air tube from the compressor to the gundrill adapter. I’m using a POS harbor freight lathe for this procedure, and really wish I had something better.
If there is any near you, you can use elder. That wood has a naturally hollow center, and has been used from time immemorial for folk instruments of all types. It is also reasonably hard. You will need a long spade bit of the right diameter, with a center bit that can follow the natural bore of the elder. You will also have to dry it first.
There ar esome other types with the holow center, but less frequent. And not all of these are hard enough.